Landau-Zener-Stueckelberg interferometry with driving fields in the quantum regime

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by  S. Ashhab

Landau-Zener-Stueckelberg interferometry with driving fields in the quantum regime
S. Ashhab1
1Qatar Environment and Energy Research Institute,
Hamad Bin Khalifa University, Qatar Foundation, Doha, Qatar
(Dated: March 21, 2017)
We analyze the dynamics of a two-level quantum system (TLS) under the influence of a strong
sinusoidal driving signal whose origin is the interaction of the two-level system with a quantum
field. In this approach the driving field is replaced by a harmonic oscillator that is either strongly
coupledtotheTLSorpopulatedwithalargenumberofphotons. StartingfromtheRabimodel,we
deriveexpressionsfortheTLS’soscillation frequenciesandcomparetheresultswiththoseobtained
7
from the model where the driving signal is treated classically. We show that in the limits of weak
1
coupling and large photon number,the well-known expression for theRabi frequency in thestrong
0
driving regime is recovered. In the opposite limit of strong coupling and small photon number, we
2
find differences between the predictions of the semiclassical and quantum models. The results of

r the quantum picture can therefore be understood as Landau-Zener-Stueckelberg interferometry in
a
thefully quantumregime.
M

9 I. INTRODUCTION couldstillobtainnon-decayingsinusoidaloscillationsbut
1
with a frequency that is completely different from what

] Landau-Zener-Stueckelberg (LZS) interferometry is the semiclassical model predicts. We shall analyze this
h encountered when a parameter of a quantum system pointandothersimilaritiesanddifferencesinthe predic-
p are varied periodically in time such that the system tions of the two models.
-
t repeatedly traverses an avoided crossing in its energy The remainder of this paper is organized as follows:
n level diagram [1]. The response of the quantum sys- in Sec. II we introduce the semiclassical and fully quan-
a
u temundersuchstrongdrivingexhibitscharacteristictwo- tum models for describing a driven two-level system. In
q dimensional interference patterns that reflect the effects Sec. III we review the case of weak driving. In Sec. IV
[ of interference involving the two traversals in a single we address the case of strong driving and LZS interfer-
driving period as well as the interference between oper- ometry,asseeninthesemiclassicalandquantummodels.
2
ations corresponding to different driving periods. This Weanalyzethe expressionsforthe Rabifrequencyinthe
v
4 situationhasbeen the subject ofnumerousstudies inre- two models and compare the two expressions. In Sec. V
7 cent years, covering both theory [2–7] and experiment we present the results of time-domain simulations of the
9 [8–19]. dynamics, giving a different perspective on the problem.
0 In the LZS problem, the driving field is generally We conclude with some final remarks in Sec. VI.
0
treatedclassically,andonlythedriventwo-levelsystemis
.
1 treated quantum mechanically. This picture can be con-
0 sidereda semiclassicalapproximationofafully quantum II. SEMICLASSICAL AND FULLY QUANTUM
7 PICTURES
treatment where the sinusoidal driving field is replaced
1
by a harmonic oscillator that contains a large number of
:
v photons and hence behaves classically. We consider a two-levelquantum system (to which we
i One can then ask the question: what happens when shallalsoreferasaqubit)drivenbyanexternalfieldwith
X
the driving field is treated quantum mechanically? In a sinusoidal time dependence. Specifically, we consider
ar this paper we use the Rabi model to address this ques- the Hamiltonian
tion. The closest that one can come to a classical signal
∆ ǫ+¯hAcos(ωt+φ )
ina quantumharmonicoscillatoris a coherentstate. In- Hˆsemiclassical = σˆx s σˆz, (1)
−2 − 2
deedweshowthatinthesemiclassicallimitwithcoherent
statescontainingalargenumberofexcitationquantathe where∆istheminimumenergygapattheavoidedcross-
predictions of the semiclassical and fully quantum mod- ing point, ǫ is the average bias point relative to the so-
els agree, although the results are described by seem- called symmetry point (ǫ = 0), A, ω and φ are, respec-
s
ingly different mathematical functions. One can expect tively, the amplitude, frequency and phase of the sinu-
that for coherent states with a small number of exci- soidal driving signal, and σˆ are qubit Pauli operators.
x,z
tation quanta the discreteness of the oscillator’s energy Whenh¯Aexceedsǫbyanamountthatislargecompared
levels and the fluctuations in photon number relative to to ∆, the coefficient of the second term in the Hamilto-
the average value start to have noticeable effects on the nianoscillatesbetweenpositive andnegativevalues such
dynamics of the driven two-level system. We find that that a sequence of Landau-Zener traversals is encoun-
these are not the only deviations that the fully quantum tered, and the physics of LZS interferometry is realized.
modelexhibitsinrelationtothesemiclassicalmodel. For The semiclassicalHamiltonian given in Eq. (1) can be
example,startingwithawell-definedphotonnumberone seen as an approximation resulting from an underlying
2
fully quantum Hamiltonian that treats both the driven lowest order that we need here, and it can be ignored.
two-level system and the driving field quantum me- Ignoring it yields the Hamiltonian:
chanically. That underlying Hamiltonian is the Jaynes-
E
Cummunigs (JC) Hamiltonian [20], which is also known Hˆ = qσ˜ +¯hωaˆ†aˆ λ˜σ˜ aˆ+aˆ† , (5)
quantum x z
as the Rabi-model Hamiltonian: − 2 −
(cid:0) (cid:1)
∆ ǫ where λ˜ =λcosθ. The angle θ here is defined exactly as
Hˆ = σˆ σˆ +¯hωaˆ†aˆ λσˆ aˆ+aˆ† , (2)
quantum x z z in the semiclassical case described in the previous para-
−2 − 2 −
graph. For small λ˜/E , we can further ignore the so-
(cid:0) (cid:1) q
wherenowω isthe frequencyofaquantumharmonicos- calledcounter-rotatingterms andobtainthe approxima-
cillator (to which we shall also refer as the cavity whose tion
excitations are photons) with annihilation and creation
E
operators aˆ and aˆ†, and λ is the qubit-cavity coupling Hˆ = qσ˜ +¯hωaˆ†aˆ λ˜ σ˜ aˆ+σ˜ aˆ† , (6)
quantum x + −
− 2 −
strength. ItshouldbenotedthatthesignsinthisHamil-
tonian were chosen such that they have a simple corre- where σ˜ = (σ˜ iσ˜ )/2 and the (cid:0)effect of thes(cid:1)e oper-
± z y
spondence with those in Eq. (1) and that our results do ators is to excite±or de-excite the qubit: σ˜ g = e
+
not depend on this particular choice. Note also that we and σ˜ e = g where g and e are th|e igrou|ndi
−
include the term proportional to σˆz, which is sometimes and exci|teid sta|teis of the |thie bare|qiubit Hamiltonian
omitted from the JC Hamiltonian. (i.e. Hˆ = E σ˜ /2).
q q x
When the cavity is treated classically, its field oper- Thegrou−ndstateoftheHamiltonianinEq.(6)isgiven
ators aˆ and aˆ† are replaced by the classical field values by g,0 , where the first and second indices specify, re-
αe−i(ωt+φq) and αei(ωt+φq), with α taken as a positive spec|tiveily, the state of the qubit and the number of ex-
real number. Given a number of photons n and noting citations in the oscillator. Apart from the ground state,
that n = aˆ†aˆ , the replacement of the quantum opera- thelowenergylevelsaregroupedintopairswithenergies
h i
torsbytheclassicalfieldvaluesgivestheamplitudeofthe n¯hω λ˜√n+1 above the groundstate energy,with cor-
classicalfieldasα=√n. Asaresult,inordertodescribe respo±nding eigenstates given by (g,n+1 e,n )/√2.
the same field intensity in the two models described by The fact that the energyeigenstat|es are qiu∓an|tumisuper-
Eqs. (1) and (2), one must set positions of this form naturally leads to Rabi-like oscil-
lations. If, for example, the system is initially set in a
4λ√n=h¯A. (3) state with the qubit in its ground state g and the cavity
in a state with n photons, and hence the combined sys-
In order to obtain fuller correspondence between the temisinitiallyinthestate g,n ,thesystemwillundergo
semiclassicalandquantumtreatments,thephasesφs and oscillations between the st|atesig,n and e,n 1 with
φq must also be set to the same value, and we shall set frequency 2λ˜√n. | i | − i
both of them to zero for simplicity here.
Obviously the two descriptions above, with oscillation
frequencies given by A/2 cosθ and 2λ√ncosθ, are es-
×
sentially equivalentandgive the same value forthe Rabi
III. RESONANT DRIVING OR COUPLING IN frequency when we set 4λ√n=h¯A.
THE WEAK LIMIT
First let us consider the simple case of weak driving, IV. STRONG DRIVING
which is the case studied by Rabi in Ref. [21]. If we set
¯hω = E (where E = √ǫ2+∆2) and we assume that We now turn to the case of strong driving. We start
q q
the drivingis weak(i.e.¯hA E ), wefindthatthe two- withthe semiclassicalmodel. Whenwesetk¯hω =ǫwith
q
≪
level quantum system undergoes Rabi oscillations with ω ∆, we obtain Rabi oscillations with frequency
frequency (A/2) cosθ, where θ =tan−1(ǫ/∆). ≫
× ∆ A
Similarly we can take the JC Hamiltonian in the case Ω = J , (7)
Rabi,s k
¯hω = E , and with a simple rotation of qubit reference ¯h ω
q (cid:18) (cid:19)
frame write it as
whereJ isthek-thorderBesselfunctionofthefirstkind
k
E [23]. This behaviour was observed recently using super-
Hˆ = qσ˜ +h¯ωaˆ†aˆ λ(cosθσ˜ +sinθσ˜ ) aˆ+aˆ† .
quantum x z x conducting qubits [24, 25]. Note that although Eq. (7)
− 2 −
(4) does not contain ǫ explicitly, this parameter is of course
(cid:0) (cid:1)
The weak-coupling regime in this model is given by the crucial in determining that the driving is resonant and
conditionλ E [22]. Inthissection,aswellasinparts therefore in determining the value of k. Note also that
q
≪
ofthefollowingsections,weassumethatthesystemisin here we consider only the case where the resonance con-
the weak-coupling regime. In this regime the part of the dition k¯hω =ǫ is satisfied, because this is the case when
coupling termin Eq.(4) thatcontains σ˜ does notaffect thequbitexhibitsthecharacteristicLZS-Rabioscillation
x
the eigenvalues or eigenstates of the Hamiltonian to the dynamics [1].
3
Ifwewanttoinvestigatethesamesituationinthefully value corresponds to the so-called deep-strong-coupling
quantum picture, we can start by writing the Hamilto- regime of the Rabi model [29–33]. The two expressions
nian in the form for the Rabi frequency are plotted in Fig. 2. The two
expressions agree at large values of n, but we now see
Hˆquantum =Hˆ0+Hˆ1, (8) cleardeviationsatsmallvaluesofn. The deviationsalso
extend to larger values of n with increasing values of k.
where
Forexample,fork =0thetwoexpressionsstarttoagree
Hˆ = ǫσˆ +¯hωaˆ†aˆ λσˆ aˆ+aˆ† very well when n > 50, whereas we need n > 500 for
0 z z
−2 − k =5. Furthermore∼, below n=10 it seems tha∼t the two
Hˆ = ∆σˆ , (cid:0) (cid:1) (9) calculations sometimes give completely different results.
1 x
−2 If we push the qubit-cavity coupling strength to even
largervalues,specificallyλ/(h¯ω)=3inFig.3,theneven
which leads to the generalizedrotating-waveapproxima-
tion described in Ref. [26]. The eigenstates of Hˆ are fork =0thetwoexpressionsfortheRabifrequencystart
0 to agree only when n > 104 (not shown in the figure).
given by
One interesting feature∼that we can see in Fig. 3 is that
λ λ for n 4 and small k the expression derived from the
Dˆ n and Dˆ n , (10) ≤
quantummodelisessentiallyzero,whilethesemiclassical
|↑i⊗ ¯hω | i |↓i⊗ −¯hω | i
(cid:18) (cid:19) (cid:18) (cid:19) derivation gives a finite value for the Rabi frequency.
with respective energies ǫ + n¯hω λ2. Here the A conclusion that we can draw from Figs. 1-3 is that
∓2 − h¯ω the semiclassical and quantum models give the same re-
displacement operator Dˆ(x) = ex(aˆ−aˆ†), and we have
sultsfortheRabifrequencyinthe limitsofsmallλ/(h¯ω)
used the state definitions σˆ = and σˆ =
z|↑i |↑i z|↓i and large n. Based on this observation, we can use
. When ǫ = k¯hω, the Hamiltonian Hˆ has de-
0 Eqs. (7) and (11) to deduce the approximation
−|↓i
generacies between the states Dˆ λ n+k and
|↑i⊗ h¯ω | i
|↓i⊗Dˆ −h¯λω |ni. These degeneracies(cid:0)are(cid:1)lifted by Hˆ1 J 4x√n e−2x2(2x)k n! Lk 4x2 , (12)
with the(cid:0)split(cid:1)tings, and hence Rabi oscillation frequen- k ≈ s(n+k)! n
cies, given by
(cid:0) (cid:1) (cid:2) (cid:3)
which is valid for small x and/or large n. It is interest-
k 2
∆ 2λ n! 2λ
Ω = e−2λ2/(h¯ω)2 Lk , ingthatconsideringthesamephysicalproblemfromtwo
Rabi,q ¯h (cid:18)¯hω(cid:19) s(n+k)! n"(cid:18)¯hω(cid:19) # differentperspectiveshasledustoinferarelationshipbe-
(11) tweentwomathematicalfunctionsthatarenotobviously
where Lk are associated Laguerre polynomials. related. Asimilarsituationisalsogivenintheappendix.
n
We can now compare the predictions of the semiclas- Itisalsoworthnotingherethatrecentlyanexpression
sicaland quantum models by comparing the two expres- containingtwo Besselfunctions was derivedforthe Rabi
sions given in Eqs. (7) and (11). An important point to frequency in the case of ǫ=0 and h¯ω =∆ [34, 35]. It is
notehereisthat,becauseoftherelationinEq.(3),there not obvious how this expressioncan be derivedfrom the
are two ways to obtain a large effective driving field in quantum model.
the quantum model, namely by having a large value of When considering the dynamics in the fully quantum
either λ orn suchthat their productis comparableto or picture, it is also interesting to consider the back-action
larger than ¯hω. ofthe qubitonthe drivingfield. Inthe semiclassicalpic-
We start with the case of weak coupling between the ture,thefieldisanexternallygivenfunctionoftimethat
qubit and the cavity (i.e. λ/(h¯ω) 1), where strong is not affected by the state of the driven system. In the
≪
driving would require a large value of n. In Fig. 1 we quantum picture, any change in the state of the qubit
plot the Rabi frequency as obtained from the semiclas- will be accompanied by a change in the state of the cav-
sical and fully quantum calculations for λ/(h¯ω) = 0.1 ity. For example, the excitation of the qubit from the
for the four cases k = 0,1,2 and 5. In fact in some re- ground to the excited state in a k-photon resonance will
cent studies, e.g. Refs. [27, 28], the value λ/(h¯ω) = 0.1 be accompaniedbythe absorptionofk photonsfromthe
has been identified as being in the ultrastrong-coupling cavity. The alert readermight have already noticed that
regime of the Rabi model. However, for purposes of this the expression that we used for the semiclassical model
study this value of λ/(h¯ω) can be considered to lie in contains the photon number n, even though the oscilla-
the weak-coupling regime because it leads to the same tions involve alternation between n and n+k photons.
behaviouras whatwe wouldobtainfor verysmallvalues Taking this point into consideration, one might think
ofλ/(h¯ω). We cansee fromFig. 1 thatthere is excellent that it would be better to set the classical field ¯hA to
agreement between the semiclassical and quantum mod- 4λ n+k/2 insteadof 4λ√n. A closerinspection of the
elsintheirpredictionoftheRabifrequencyforallvalues functionsplottedinFigs.1-3revealsthatthesituationis
p
of n, at least up to k =5. somewhat more complicated. For λ/(h¯ω) = 0.1, taking
Next we consider the case of strong coupling between largevaluesofnwefindthatweobtaintheclosestagree-
the cavity and the qubit, and we set λ/(h¯ω) = 1. This mentbetweenthesemiclassicalandquantumcalculations
4
bysetting¯hA=4λ n+k/2+0.5forallfourvaluesofk semiclassicalmodel. When we extend the simulations to
plotted in Fig. 1. The term k/2 is therefore consistently longer timescales, we see a decay in the oscillations with
p
there. However, we also have the additional 0.5 whose adecaytimeof 30 2π/Ω . Thisvalueforthedecay
Rabi
∼ ×
originis not clear. Fromfitting the data for all the com- time isratherlargeconsideringthatthe quantumfluctu-
binations of λ/(h¯ω) 0.1,1,2,3 and k 0,1,2,5 ations in n in such a coherent state are √1000, which is
∈ { } ∈ { }
we consistently find that the best agreement is obtained about 3% of the mean photon number. The reason for
whenwe set h¯A=4λ n+k/2+0.5 [λ/(h¯ω)]2/3. We this weak decay is that the point A/ω =10 is very close
−
do not know the origin of the last two terms inside the to a maximum in J (A/ω) and fluctuations up to 3% in
2
p
square-root. We also note that for small values of n this A do not result in large fluctuations in Ω .
Rabi
formulagavegoodagreementfor smallvalues ofλ/(h¯ω), Whenwechangetheparameterssuchthatwenowhave
but deviations between the two expressions persisted, 100photons inthe coherentstate, we seeveryfastde-
∼
especially for λ/(h¯ω) 2, where no value of the shift cay in the oscillations, with a decay time of only a few
≥
seemed to consistently reduce the deviations. times the Rabi oscillation period. The quantum fluctua-
Anotherback-actioneffectarisesnaturallyif foramo- tionsinnarenowontheorderof10%,whichmeansthat
mentweconsiderwhathappenswhenwechooseparame- the Rabi frequency varies significantly for the different
tersthatdonotsatisfyresonanceconditions. Inthiscase Fock states that make up this coherent state. Interest-
the energy eigenstates are divided into two groups cor- ingly,ifweuseaFockstatewithexactly100photonsand
responding to the qubit states and , as described the coupling strength adjusted to the appropriate value,
|↑i |↓i
by Eq. (10). The cavity part of each one of these en- we recover decay-less oscillations with a frequency that
ergy eigenstate is described by a Fock state, just as in is essentially identical to that obtained from the semi-
an isolated harmonic oscillator, but with a qubit-state- classical calculation. This result is somewhat surprising,
dependent displacement. If the qubit is prepared in one becausecoherentstatesaregenerallyexpectedtogivethe
of the states and and the cavity is initially pre- closest similarity to classical fields.
|↑i |↓i
pared in a coherent state, then the cavity field will un- Whenwereducetheaveragenumberofphotonsinthe
dergo oscillations with frequency ω about a qubit-state- coherent state to 10, the decay becomes so fast that we
dependent equilibriumpoint that is shifted fromthe ori- can barely see the Rabi oscillations, and instead of the
gin by λ/(h¯ω). Even if we choose parameters that sat- step-like dynamics that is characteristic of LZS interfer-
±
isfy a resonance condition, these oscillations will occur ometry we now see only a few frequency components in
on short timescales. On longer timescales, specifically the dynamics.
the timescale of Rabi oscillations,the qubit will undergo
oscillationsbetweenthestates and ,andthecavity
|↑i |↓i
fieldwilladjustitsoscillationpatternsuchthatitremains VI. CONCLUSION
correlatedwiththequbit’sstate. Forexample,afterafull
transfer of probability from the state to the state
|↑i |↓i We have treated the problem of Landau-Zener-
or vice versa the cavity field will have shifted the origin
Stueckelberg interferometry when the driving field is in
of its oscillations by 2λ/(h¯ω). As discussed above, the
the quantum regime. We have found that the quan-
amplitude of the cavity field oscillations will also change
tum treatment reproduces the results of the semiclas-
because of the absorption or emission of k photons in a
sical treatment when the qubit-cavity coupling strength
k-photon resonance.
is small or the number of photons in the cavity is large.
In this case the expressions containing Laguerre polyno-
mials that are typical in the study of the Rabi model
V. TIME-DOMAIN SIMULATIONS coincide with the expressions that contain Bessel func-
tions and are typical in strong-driving problems. The
We now present the results of simulations where we semiclassicalexpressionis no longerapplicable,however,
solvetheSchr¨odingerequationandobtainthequbitstate for large values of λ/(h¯ω) and small values of n. In this
populations as functions of time in the semiclassical and case, not only do quantum fluctuations cause decaying
fully quantum models. We take the case of two-photon oscillationsin the time-domain, but the n dependence of
resonance ǫ = 2h¯ω. We choose a driving amplitude the Rabi frequency itself shows differences in the predic-
A/ω = 10 (or an equivalent value of n in the quantum tions of the two models.
model), which gives constructive interference and there- InoursimulationsoftheRabimodelwehavereliedon
foreareasonablyhighRabifrequency. Forthe qubitgap numerical diagonalization of the Hamiltonian. It should
we choose the value ∆/(h¯ω) = 0.4, which is relatively be noted that the recent advances made on the integra-
large and clearly shows the step-like dynamics that is bility and solution of the Rabi model provide additional
one of the characteristics of LZS interferometry [17]. analytical tools and allow the simulation of very large
The results of the simulations are plotted in Fig. 4. systemusingreasonablecomputationalresources[36–39].
When we use the quantum model with a coherent state These new tools might be useful for treating problems
that contains 1000 photons, the results of the quan- similar to the ones considered here. For example, the
∼
tummodelarealmostindistinguishablefromthoseofthe new techniques developed in Refs. [36–39] could make it
5
possibletoanalyzethecaseofslow-passageLZSinterfer- ∆ is given by
ometry, which would correspond to very small values of
¯hω/∆ and is therefore challenging for us to treat using ∆ A
Ω = J . (13)
our numerical approach. Rabi ¯h k ω
(cid:18) (cid:19)
As there is continuing effort to increase the coupling
strength in cavity-QED systems with remarkable recent This expression is expected to be valid for all values of
progress and as driving quantum systems with oscillat- A/ω. Thereisawell-knownapproximationfortheBessel
ingfields isone ofthe maintoolsofquantumcontrol,we function Jk(x) that is valid in the limit x k, namely
≫
expect that the present study combining these two im-
portant problems will be relevant to future studies that 2 π
J (x) cos x (2k+1) , (14)
pushthelimitsofLandau-Zener-Stueckelberginterferom- k ≈ πx − 4
r
etry and cavity QED. (cid:16) (cid:17)
We would like to thank S. Shevchenko for useful dis- which gives the approximation
cussions.
∆ 2ω A π
Ω cos (2k+1) . (15)
Rabi
≈ ¯h πA ω − 4
r (cid:18) (cid:19)
Appendix A: Approximation for Bessel functions
obtained from LZS interferometry Using a calculation based on the adiabatic-impulse
picture, where one constructs the dynamics using well-
In this appendix we describe an approximation for known expressions for the mixing of probability am-
Besselfunctions that arises naturally fromanalyzing the plitudes at avoided crossings and the accumulation of
problem of LZS interferometry. phases away from avoided crossings, Ref. [6] derived an
As mentioned in the main text, the Rabi frequency expressionfortheRabifrequencythatisvalidinthelimit
when strongly driving a k-photon resonance with ¯hω ¯h(A ǫ)/∆ 1 and ¯h2√A2 ǫ2ω/∆2 1:
≫ − ≫ − ≫
∆ 2ω A2 (kω)2 kω π
Ω = cos − kcos−1 , (16)
Rabi
¯hsπ A2 (kω)2 p ω − A − 4!
−
p
where we have used the k-photon resonance condition k 1,whichislessstringentthantheconditionx/k 1
≫ ≫
ǫ = k¯hω. This expression differs from the one given required for the standard approximation for the Bessel
in Eq. (15) by the fact that instead of A we now have function. Using numerical calculations we have verified
A2 (kω)2 and we now have the arccosine function that this is indeed the case. In fact, we find that the
−
for one of the phases inside the cosine. These differences approximation is very good almost down to the point
p
disappear when we take the limit A/(kω) 1. How- x=k.
≫
ever, the expression derived using the adiabatic-impulse We also find that the further approximationwhere we
method does not require the condition A/(kω) 1, but expand each term inside the cosine function to next-to-
≫
ratherthe lessstringentcondition¯h(A kω)/∆ 1. As leading order in k/x, i.e.
− ≫
a result, the expression should remain valid even away
from the limit A (kω). By comparing Eqs. (13) and 2 π π k2
(16) we can deduc≫e the approximation Jk(x) cos x k + , (18)
≈sπ√x2 k2 − 2 − 4 2x
− (cid:18) (cid:19)
J (x) 2 cos x2 k2 kcos−1 k π . isagoodapproximationalmostdowntox=kfork<20.
k
≈sπ√x2 k2 − − x − 4 We have therefore derived a mathematical relatio∼n by
− (cid:18)p (cid:19)
(17) lookingataphysicalproblemfromtwodifferentperspec-
This approximation should remain valid as long as x tives.
−
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p
7
1
(a)
0.5
∆
/bi 0
Ra
Ω
-0.5
-1
0 5 10 50 100 500 1000
n
1
(b)
0.5
∆
/bi 0
Ra
Ω
-0.5
-1
0 5 10 50 100 500 1000
n
1
(c)
0.5
∆
/bi 0
Ra
Ω
-0.5
-1
0 5 10 50 100 500 1000
n
1
(d)
0.5
∆
/bi 0
Ra
Ω
-0.5
-1
0 5 10 50 100 500 1000
n
FIG. 1: Rabi frequencies obtained from the semiclassical
model (squares, calculated using Eq. 7) and from the fully
quantummodel(+signs,calculatedusingEq.11)asfunctions
of photon number n. In all the panels we set λ/(¯hω) = 0.1,
which corresponds to the weak-coupling regime. The differ-
entpanelscorrespondtodifferentvaluesofkintheresonance
condition k¯hω = ǫ. In particular we take the cases k = 0
(a), 1 (b), 2 (c) and 5 (d). The x axis is divided into three
parts that have different scales: the first part ranging from
0 to 10, then second from 10 to 100 and the third from 100
to1000. Wenoteherethatthefrequencyshouldbeobtained
by taking the absolute value of the relevant expressions, but
we keep the signs here in order to capture what can be con-
8
0.5
(a)
∆
/Rabi 0
Ω
-0.5
0 5 10 50 100 500 1000
n
0.5
(b)
∆
/Rabi 0
Ω
-0.5
0 5 10 50 100 500 1000
n
0.5
(c)
∆
/Rabi 0
Ω
-0.5
0 5 10 50 100 500 1000
n
0.5
(d)
∆
/Rabi 0
Ω
-0.5
0 5 10 50 100 500 1000
n
FIG.2: SameasinFig.1,butwithλ/(¯hω)=1,whichcorre-
sponds to thestrong-coupling regime.
9
0.25
(a)
∆
/Rabi 0
Ω
-0.25
0 5 10 50 100 500 1000
n
0.25
(b)
∆
/Rabi 0
Ω
-0.25
0 5 10 50 100 500 1000
n
0.25
(c)
∆
/Rabi 0
Ω
-0.25
0 5 10 50 100 500 1000
n
0.25
(d)
∆
/Rabi 0
Ω
-0.25
0 5 10 50 100 500 1000
n
FIG.3: SameasinFig.1,butwithλ/(¯hω)=3,whichcorre-
sponds to extremely strong coupling.
10
1
0.5
P
0
-0.5
0 5 10 15 20
ωt/(2π)
FIG. 4: Occupation probability P↓ of the qubit state |↓i as
a function of time t with driving amplitude A/ω =10, qubit
bias ǫ/(h¯ω)=2 and qubitgap ∆/(¯hω)=0.4. The red line is
obtainedfromthesemiclassical model. Thegreenline,which
isshifteddownby0.25inordertomakeitresolvablefromthe
redline,isobtainedfromthequantummodelwiththecavity
initially set in a coherent state with hni = 1000, and the
couplingstrengthλischosensuchthatn=1000corresponds
to A/ω = 10. There red and green line agree very well and
would hardly be resolvable without the shift. The blue line,
which is shifted down by 0.5, is obtained from the quantum
model with hni = 100. It shows a clear deviation from the
othertwolines. Inparticular, it exhibitsa decaythatcan be
attributedto thesignificant spread in theRabifrequencyfor
thedifferentnvaluesinthecoherentstate. Themagentaline,
which is shifted down by 0.75, is obtained from thequantum
model with hni = 10. Instead of the step-like dynamics, we
can now see that there are only a few frequency components
inthedynamics,aswould beexpectedfor suchasmall value
of hni.