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Quantum spectroscopy with Schrödinger-cat statesSemiconductor quantum spectroscopy with Schrodinger cat states

M. Kira,1,⋆, S.W. Koch,1 R.P. Smith,2 A.E. Hunter,2 and S.T. Cundiff,21Department of Physics, Philipps University Marburg, Renthof 5, D-35032 Marburg, Germany and

2JILA, University of Colorado and National Institute of Standards and Technology, Boulder, CO 80309-0440, USA

(Dated: July 14, 2011)

We apply the central principles of quantum mechanics to rigorously extend classical laser spec-troscopy to the realm of quantum-optical spectroscopy. Based on the fundamental quantum-opticalproperties [1–11], we throughly discuss which generalizations are needed to go beyond classical laserspectroscopy. We also present the details of our experiments and determine the explicit form ofthe cluster-expansion transformation (CET) and other important aspects and generalizations of ourapproach. The convergence and noise-filtering capabilities of the CET are also benchmarked.

Contents

I. Quantum-mechanics foundations 1A. Properties of the density matrix 2B. Linearity of quantum mechanics 3C. Classical vs. quantum averaging 4

II. From classical to quantum description of single-mode lasers 5A. Heisenberg uncertainty principle 6B. Coherent states 7C. Glauber-Sudarshan function 8D. Classical vs. quantum sources 8

III. From classical to quantum-optical spectroscopy 9A. Differential spectroscopy 10B. Quantum-optical responses using the cluster-expansion transformation 12C. CET reconstruction of quantum-Rabi fopping 14D. Atomic vs. quantum-well absorption spectroscopy 16

IV. Projected quantum-optical responses to semiclassical and quantum sources 17A. Thermal source 17B. Squeezed vacuum 17C. Schrodinger cat states 18D. Constructing N -photon increment perturbations for the laser 20E. General projection of the system response to Schrodinger cat states 22

V. Quantum-optical spectroscopy with semiconductor quantum wells 22A. Experimental setup 22B. Squeezing- and stretching-cat perturbations 24C. Convergence of the CET projection 25D. Noise sensitivity in CET projection 26E. Squeezing dependence of the differential signal 27

References 28

I. QUANTUM-MECHANICS FOUNDATIONS

Quantum mechanically, any system is defined by its Hamiltonian H. However, for the general purposes of quantum-optical spectroscopy, we do not need to explicitly specify the system studied. Instead, we only need to know that itsdensity matrix, ρ(t), evolves according to the Liouville-von Neumann equation [12]:

i∂

∂tρ(t) =

[

H, ρ(t)]

−= Hρ(t)− ρ(t)H , (1)

© 2011 Macmillan Publishers Limited. All rights reserved.

2

where the “hat” denotes the quantum-mechanical operator nature of both H and ρ(t). 0 We add the ”hat” to anoperator only when it is needed to clearly mark the operator nature of the particular quantity.

If the system is coupled to a reservoir, one can add the Lindbladian [13]

i~∂

∂tρ(t)

∣

∣

∣

∣

L

= iL [ρ(t)] ≡ i∑

λ,ν

γλ,ν

[

L†λLν ρ(t)− 2Lν ρ(t) L

†λ + ρ(t) L†

λLν

]

, (2)

to the right hand-side of Eq. (1). The appearing Lλ and L†λ are annihilation and creation operators, respectively,

within the system studied. The Lindblad formulation is needed only if our knowledge about the system is incomplete,i.e., if the system is not closed and we are not able to describe all elements of the “outside” world microscopically.We will use the fact that the linearity of quantum mechanics is preserved for both open and closed systems regardlesshow complicated H and the reservoir coupling γλ,ν are.

A. Properties of the density matrix

Before analyzing the consequences of Eqs. (1)–(2), we make a strict connection between the quantum theory andmeasurements recording the quantity R, such as the excitation level of the system studied. In particular, we considerthe situation where the individual measurements are repeated after the system has been re-initialized to the sameinitial conditions. Even though we can easily identify the operator R corresponding to the measured quantities,the quantum theory does not determine the outcomes Rj of the individual measurements. Instead, the quantummechanical observable average is connected with the ensemble average of the measurements:

〈R〉 ≡ Tr[

R ρ(t)]

= Tr[

ρ(t) R]

= limN→∞

1

NN∑

j=1

Rj , (3)

where Rj is the j-th outcome of measurements repeated N times. Since ρ(t) is same for all possible R and we can

choose R arbitrarily, ρ(t) must necessarily define all quantum properties of the system. Therefore, ρ(t) provides onepossible representation of the quantum statistics that defines all fluctuation properties of the system.

By choosing R = I, we see that the density matrix must satisfy the norm

〈I〉 = Tr[

I ρ(t)]

= Tr [ρ(t)] = limN→∞

1

NN∑

j=1

1 = 1 ⇒ Tr [ρ(t)] = 1 (4)

for all times. We also know that all measurements yield well-behaving and real-valued outcomes Rj . Therefore, both

R and ρ must be Hermitian, i.e.

ρ†(t) = ρ(t) , (5)

for all times.Any Hermitian operator has eigenstates

ρ(t)|φλ(t)〉 = pλ|φλ(t)〉 (6)

that form an orthogonal and complete set

〈φν(t)|φλ(t)〉 = δν,λ ,∑

λ|φλ(t)〉〈φλ(t)| = I . (7)

Therefore, we always can find a basis that diagonalizes the density matrix via

ρ(t) = Iρ(t)I =∑

ν, λ

|φν(t)〉〈φν(t)| ρ(t) |φλ(t)〉〈φλ(t)| =∑

ν,λ

pλ(t)|φν(t)〉〈φν(t)|φλ(t)〉〈φλ(t)|

=∑

λ

pλ(t) |φλ(t)〉〈φλ(t)| (8)

that follows directly from the properties (7).We can now insert the result (8) into Eq. (3) yielding


3

〈R〉 = Tr

[

R∑

λ

pλ(t) |φλ(t)〉〈φλ(t)|]

=∑

λ

pλ(t) Tr[

R |φλ(t)〉〈φλ(t)|]

≡∑

λ

pλ(t) 〈Rλ(t)〉 , (9)

where 〈Rλ(t)〉 ≡ 〈φλ(t)|R|φλ(t)〉 defines the expectation value for the quantity R when the system is in the state

|φλ(t)〉. Therefore, 〈R〉 can always be represented via an analogous decomposition as the density matrix. By choosing

an identity R = I and a projection measurement R = |φν(t)〉〈φν(t)| in Eq. (9), we find that the density matrix mustsatisfy the conditions of normalization and positivity:

∑

λpλ = 1 , pν ≥ 0 , ∀ ν (10)

for any physical ρ(t). Therefore, we find the standard probabilistic interpretation where pν defines the probability offinding the system in the state |φν(t)〉. Here, pν identifies a positive-definite metric to project Rν representations ofoperator outcomes whenever ρ is decomposed into orthogonal states. We discuss in Sec. I C that this property has farreaching implications for averaging procedures.

B. Linearity of quantum mechanics

In many situations, it is meaningful to decompose the initial density matrix of the system according to

ρ(0) =∑

βpβ ρβ , Tr [ρβ ρβ′ ] 6= δβ,β′ for some β 6= β′ (11)

which turns into Eq. (8) if we set ρβ → |φλ(t)〉〈φλ(t)|. For the sake of generality, we do not apply this substitutionbut consider a generic decomposition where each individual state ρβ is a physical density matrix. This more generaldecomposition does not necessarily satisfy the positivity (10) because it does not fulfill the orthogonality of the basisstates. In other words, when the ρβ are not orthogonal, we cannot assign a probabilistic interpretation to pβ definingthe weights of the ρβ components.To reveal the fundamental implications of the expansion (11), we only impose the constraints

∑

βpβ = 1 , pβ ∈ R (12)

which imply the normalizability of pβ but not its positivity such that pβ is a quasiprobability distribution. Thisseemingly harmless modification will generate extraordinary projection capabilities between the ρβ states and arbitraryρ(0). These features are discussed in Sec. II where we show how classical laser spectroscopy, performed with anonorthogonal ρβ basis, can be projected into quantum-optical spectroscopy defined by the generic ρ(0).

Since each individual ρβ component is still a physical density matrix, its time evolution follows from

i~∂

∂tρβ(t) = H ρβ(t)− ρβ(t) H + i

∑

λ,ν

γλ,ν

[

L†λLν ρβ(t)− 2Lν ρβ(t) L

†λ + ρβ(t) L

†λLν

]

, (13)

as we combine the Liouville-von Neumann Eq. (1) and the Lindblad Eq. (2). This constitutes a fully linear equationregardless how complicated the system Hamiltonian is. Therefore, the individual ρβ(t) solutions construct the fullquantum dynamics via the same transformation as used for the initial state, i.e.,

ρ(t) =∑

β pβ ρβ(t) (14)

where pβ is defined by the initial conditions. Inserting this expression into Eq. (3) produces

〈R(t)〉 = Tr[

R∑

βpβ ρβ(t)]

=∑

βpβTr[

R ρβ(t)]

≡ ∑

βpβ 〈Rβ(t)〉 . (15)


4

where 〈Rβ(t)〉 is the average for a situation where ρβ(0) is the initial state of the system. We see that the average of anyobservable follows from the same linear-decomposition relation (11) as the initial density matrix. This fundamentalproperty follows directly from the linearity of quantum mechanics, i.e. the linearity of the evolution (13). This featureis the key point in defining the connection between classical and quantum-optical spectroscopy, as shown in Sec. II.

a “Classical” Rbb “Quantum” < >R

probedsystem

probedsystem

r1

r2

r3

r4 r

5

r( )t

FIG. 1: Schematic comparison of set of, a, classical measurements vs., b, a single quantum measurement. The black arrowsindicate the individual classical measurements performed initializing the system to the states ρβ ; the shaded area symbolizes the probedsystem. The quantum measurement (shaded arrow with structure) can alternatively be constructed using the decomposition (14) into all

possible classical initial states.

We often have the situation that the specific system property 〈Rβ(t)〉 of interest is easiest to detect using thespecific experimental initialization of the states ρβ . In other words, it is convenient to generate a given set ρβ asinitial states for the measurements whereas other initial conditions are tedious to generate; explicit examples follow inSecs. II–V. The collected set of 〈Rβ(t)〉 outcomes can then be used to project the outcome 〈R(t)〉 resulting from anyarbitrary initial state ρ =

∑

β pβ ρβ , provided that this decomposition covers all possible initial-state configurations.This projection becomes extremely useful when the generic initialization of ρ is complicated whereas the states ρβ canbe accessed relatively easily. In the Article, we have used this concept to construct the scheme of quantum-opticalspectroscopy from a set of measurements provided by classical laser spectroscopy. Therefore, we need to establish ascheme similar to Eq. (11) to define the relation between these spectroscopies. We schematically illustrate our generalconcept in Fig. 1; the explicit formulation follows in Sec. III.

C. Classical vs. quantum averaging

In classical mechanics, the system properties contain no fluctuations if the initial conditions are known precisely.A single initial condition β yields a precise classical outcome Rcl

β (t) at the later time t. Therefore, fluctuations in aclassical system must stem for initial conditions that are known only with limited accuracy. In this situation, it isconvenient to introduce a probability distribution pclβ for the initial states. If we examine the system properties at alater time, we then find an average outcome

Rcl(t) =∑

βpclβ Rcl

β (t) (16)

that defines the classical averaging procedure where pclβ is a normalized and positive-definite distribution of initialstates.

We notice that the quantum-mechanical Eq. (15) has an analogous form as Eq. (16) if we limit pβ to be positivedefinite. Therefore, it is interesting to study whether classical averaging can generate new unexpected properties. Forthis purpose, we analyze what happens to the expectation values of a fluctuation operator of the position

R→ ∆x2 = (x− 〈x〉)2 , (17)

when the classical averaging is applied. As a general property, the fluctuations ∆x2 must always be positive for anystate ρβ .[14, 15]

To implement classical averaging, we use Eq. (15) with the constraint pβ ≥ 0, yielding

〈∆x2〉 = ∑

βpβ 〈∆x2β(t)〉 ≤∑

βpβ 〈∆x2max〉 = 〈∆x2max〉 , 〈∆x2〉 ≥ ∑

βpβ 〈∆x2min〉 = 〈∆x2min〉 (18)

that follows directly from the imposed positivity of the metric and norm, Eq. (12). Furthermore, we substitutedthe individual fluctuations either by 〈∆x2max〉 or 〈∆x2min〉 as the maximum or the minimum value of the fluctuations


5

among the states ρβ , respectively. We see that the classical averaging produces a strict condition

〈∆x2min〉 ≤ 〈∆x2〉 ≤ 〈∆x2max〉 . (19)

This means that the classical averaging cannot change the system properties beyond those appearing for the individualρβ states. Consequently, the classical averaging includes fluctuations only in the probabilistic sense while interestingnew quantum phenomena — such as squeezing of 〈∆x2〉 below the classically observed value — must stem from effectsbeyond classical averaging.

To illustrate these new aspects, we analyze how wave-like interference signatures — that can be considered as theelementary quantum features — can be generated. For this purpose, we analyze an alternative form of “adding”states together via the superposition principle:

|ψ〉 = ∑

λ cλ |φλ〉 , 〈φλ|φν〉 = δλ,ν , 〈ψ|ψ〉 = 1 ,∑

λ|cλ|2 = 1 , (20)

where we have summed over the wave functions instead of the density matrices. It is customary to choose the states|φλ〉 orthogonal with respect to each other, yielding a simple normalization relation. This |ψ〉 is always a physicallyallowed state and the corresponding density matrix becomes

ρQM ≡ |ψ〉〈ψ| = ∑

λ,λ′ cλ c⋆λ′ |φλ〉〈φλ′ | = ∑

β |cλ|2 ρλ +∑

λ,λ′ 6=λ cλ c⋆λ′ |φλ〉〈φλ′ | , (21)

where ρλ ≡ |φλ〉〈φλ| is a density matrix for the state λ. We see that only the diagonal part can be expressed in aform that is compatible with Eqs. (11) and (16). The remaining nondiagonal contributions provide a new class ofstates not included by classical averaging.

To see this explicitly, we consider a simple example where we apply Eq. (20) using two orthogonal momentumeigenstates 〈x|β ≡ ±k〉 = 1√

Le±ikx with quantization length L and equal weights c± = 1√

2. A straightforward

substitution converts Eq. (21) into

ρQM =|k〉〈k|+ | − k〉〈−k|

2+

|k〉〈−k|+ | − k〉〈k|2

. (22)

If we now monitor the position, i.e. R = |x〉〈x|, the detection yields

〈|x〉〈x|〉 = 12

(

|〈x|k〉|2 + |〈x|−k〉|2)

+ 12 (〈x|k〉〈−k|x〉+ 〈x|−k〉〈k|x〉) = 1

L + 1L cos 2kx, (23)

where the first part gives a constant contribution due to the “diagonal” classical averaging. This part has exactly thesame format as that resulting from either one of the states | ± k〉 alone. Therefore, the classical averaging does notproduce any new effects.

However, the superposition averaging yields a new oscillatory interference pattern that cannot be understood viaclassical averaging of momentum states. Therefore, the nondiagonal parts of ρQM can yield new quantum effects whichare directly associated with the wave properties of quantum mechanics. In this particular example, we have used asingle-particle basis. For multi-particle systems, the interference pattern can also be associated with correlations andentanglement effects.

At this state, we conclude that classical measurements cannot be extended into the quantum realm through classicalaveraging (16). However, as we discuss in Sec. II, classical laser spectroscopy operates with basis states |β〉 that are notorthogonal but overcomplete. This allows us to utilize the properties of the more general decomposition (11)–(12) andits consequences. This dramatically enhances the transformation capabilities because one can then make Eqs. (11)–(12) and (21) equivalent. To realize this projection capability in laser spectroscopy, we discsuss in Sec. II a quasiprobability distribution pβ → P (β) that projects any quantum-optical response out of a set of classical measurements.

II. FROM CLASSICAL TO QUANTUM DESCRIPTION OF SINGLE-MODE LASERS

Current laser spectroscopy benefits from sophisticated techniques to generate extremely high-intensity laser sourceswith well controlled spectral, temporal, and/or phase aspects.[16–18] In particular, one can reach a superb measure-ment accuracy by using lasers with spatial, spectral, or temporal single-mode properties. Figure 2a and 2b presentschematic examples of a spectral single-mode laser and a single-mode laser pulse, respectively. Both of them have


6

a well-defined amplitude and phase that can conveniently be described in an equivalent phase-space representationshown in Fig. 2c. Formally, the single-mode laser properties follow from the decomposition

E(r, t) = E[

u(r, t)B + u⋆(r, t)B†] (24)

where u(r, t) is the spatio-temporal mode and E defines the so-called vacuum-field amplitude.

Frequency

Inte

nsity

Spectralmode Time

Temporalmode

Ele

ctr

ic fie

ld

x quadrature

yquadra

ture ( , )x y

a b c

FIG. 2: Representations of single-mode lasers. Lasers with a single, a, frequency resonance and, b, temporal mode are shown. c,Both of them can be represented in phase space. Classically, the laser emission corresponds to an accurate (x, y) position (arrow) in phasespace. The light quantization modifies this into a phase-space distribution (shaded rings).

In the classical description, B is just a complex-valued amplitude x+ iy that produces a single point in the phase-space, indicated by the arrow in Fig. 2c. In the quantum description, one additionally takes into account that x andy are complementary operators and introduces the Bosonic creation and annihilation operators B† and B. Theseoperators

B = x+ iy , B† = x− iy (25)

can also be expressed through the quadrature operators x and y. Here, the association of x and y with positionand momentum coordinates is purely formal because they define the amplitude of the laser field rather than actualproperties of a particle. Therefore, x and y are commonly referred to as complementary quadratures of the field.

The light quantization replaces a classically accurate (x, y) point by a phase-space distribution as illustrated viathe fuzzy rings around the classical central value. The explicit form of the phase-space distributions is addressed inSec. II C. The actual quantization step follows from the introduction of canonical commutation relations [3],

[x, y]− = i2 , [x, x]− = 0 = [y, y]− (26)

or equivalently from the Bosonic commutation relations[

B, B†]− = 1 , [B, B]− = 0 =

[

B†, B†]− . (27)

The quantum nature of light implies that it is described through its quantum-statistics, i.e. a generic distribution suchas the density matrix ρ or the phase-space distributions that also determine the fundamental fluctuations. Quantum-optics investigations focus on revealing how the quantum statistical aspects emerge and what are their characteristicsignatures in the light-matter interaction. A more detailed discussion of light quantization can be found in severaltext books.[1–11]

A. Heisenberg uncertainty principle

In any laser spectroscopy, the fundamental accuracy limits are determined by the Heisenberg uncertainty relation

∆x∆y ≥ 12 |〈[x, y]−〉| = 1

4 , ∆x2 ≡ 〈x2〉 − 〈x〉2 , ∆y2 ≡ 〈y2〉 − 〈y〉2 , (28)

where Eq. (26) is applied. Due to this strict condition, light cannot be represented as a precise point in phase space as ina classical description. Instead, the light contains inherent quantum fluctuations that can be utilized when quantum-optical spectroscopy is pursued. The most classical field has minimal fluctuations ∆xcoh = 1

2 in both quadraturedirections while only true quantum states can exhibit quadrature squeezing below this minimum-uncertainty limit.

The quadrature fluctuations can be directly evaluated using the photon operators x = B+B†

2 and y = B−B†

2i . Adirect substitution into Eq. (28) produces


7

∆x2 = ∆x2coh + 12Re

[

∆〈B†B〉+∆〈BB〉]

∆y2 = ∆x2coh + 12Re

[

∆〈B†B〉 −∆〈BB〉]

,

∆〈B†B〉 ≡ 〈B†B〉 − 〈B†〉〈B〉

∆〈BB〉 ≡ 〈BB〉 − 〈B〉〈B〉. (29)

The situation where the appearing correlations in photon number and two-photon absorption vanish realizes the limitof minimal fluctuations in both quadrature directions. Therefore, both squeezing and stretching of the field requiresthe presence of photon correlations. We show in our Article that these correlation aspects grant us direct access toquantum-optical spectroscopy in many-body systems.

B. Coherent states

A perfect single-mode laser is described by a coherent state [3]

|β〉 ≡ D(β)|0〉 , D(β) ≡ eβB†−β⋆B , (30)

where |0〉 is the vacuum state, D(β) is the displacement operator, and the complex-valued β = β1 + iβ2 identifiesthe displacement amplitude. The real-space representation, density matrix, and Wigner function [19] of the coherentstate are [3–5, 10]

〈x|β〉 =

(

2

π

)14

e−(x−β1)2

eiβ2(2x−β1) , ρ|β〉 ≡ |β〉〈β|

W|β〉(x, y) ≡ 1

π

∫

dr〈x− r2 |ρ|β〉|x+ r

2 〉 e2iyr =2

πe−2(x−β1)

2

e−2(y−β2)2

, (31)

respectively. We see that the coherent state has Gaussian fluctuations in the (x, y) phase space. It is straightforwardto verify that W|β〉(x, y) produces the minimal classical fluctuations ∆x = ∆y = ∆xcoh = 1

2 in both quadraturedirections.

The coherent states produce a classical factorization

〈[B†]JBK〉 = [β⋆]JβK , ⇒ B → β, B† → β⋆ (32)

for any normally ordered expectation value. In other words, the coherent states produce a 〈[B†]JBK〉 that follows froma classical substitution of Boson operators by complex-valued amplitudes. Based on these properties, we can considercoherent states as the most classical states allowed by a quantum-mechanical description. High-quality single-modelasers approach this limit with respect to a well-defined reference.[20]

Despite exhibiting the discussed classical features, coherent states still have some quantum aspects. For example,the classical factorization (32) works only for normally ordered expectation values, which indicates that generic

expectation values such as the moments of x = B†+B2 quadrature always exhibit nonvanishing fluctuations.[3, 21] In

other words, the coherent state cannot have absolute accuracy in both quadratures, as dictated by the Heisenberguncertainty principle (28). In addition, coherent states are not orthogonal but overcomplete because they satisfy [3]

〈β′|β〉 = e−12 |β−β′|2 e−iIm[β′β⋆] ,

1

π

∫

d2β |β〉〈β| = I . (33)

As a result, one can anticipate that classical spectroscopy contains the additional information of true quantum opticsbecause one must apply the general form (11)–(12) instead of the classical averaging (16) when projecting newresponses out of a given set of classical measurements.


8

C. Glauber-Sudarshan function

The quantum statistics of light can be represented in different equivalent forms such as using the Wigner functionor the density matrix. This quantum-statistical pluralism can be utilized to enhance the visibility of certain quantum-optical aspects. Classical vs. quantum properties of light can be seen very transparently when one uses the Glauber-Sudarshan P (β) function [22, 23] that is related to the Wigner function via a Gaussian convolution [3],

W (α) =2

π

∫

d2β P (β) e−2|α−β|2 =

∫

d2β P (β)W|β〉(α) . (34)

Here, we identified the Gaussian through the Wigner function (31) of a coherent state.The inverse Fourier transformation of Eq. (31) can be applied to produce the x-space representation of any density

matrix,

〈x|ρlight|x′〉 =

∫ +∞

−∞dyW (x+x′

2 + iy) e−2i(x−x′)y =

∫

d2β P (β)

∫ +∞

−∞dyW|β〉(

x+x′

2 + iy) e−2i(x−x′)y , (35)

where we inserted the relation (34) in the last step. We see that the remaining Fourier transformation produces thedensity matrix ρ|β〉 ≡ |β〉〈β|. Therefore, the density matrix of an arbitrary quantum field becomes

〈x|ρlight|x′〉 =∫

d2β P (β)〈x|ρ|β〉|x′〉 = 〈x|∫

d2β P (β)|β〉〈β| |x′〉 = 〈x|∫

d2β P (β) ρ|β〉 |x′〉 . (36)

Since this identification works for any arbitrary light state, we must necessarily have

ρlight =

∫

d2β P (β) |β〉〈β| =∫

d2β P (β) ρ|β〉 . (37)

To the best of our knowledge this remarkable relation was first introduced independently by R.J. Glauber [22] andE. Sudarshan [23] to represent arbitrary quantum states in terms of fields that have the most classical behavior. Werecognize that the quantum statistics of all light states can indeed be expressed in the form of a generalized classicalaveraging (11) described by the quasiprobality (12) with the association P (α) → pβ . Therefore, we can convertmeasurements performed with a coherent-state laser into the responses of any quantum source, see Sec. III.

D. Classical vs. quantum sources

Before we proceed further, we first summarize some of the central features of P (β). As a general property, P (β)produces all the normally-ordered expectation via a phase-space average [22]

IJK ≡ 〈[B†]JBK〉 =∫

d2β [β⋆]JβK P (β) (38)

that is directly connected to the classical factorization (32). It is also convenient to identify the normally orderedcharacteristic function

χN (η) ≡∞∑

J,K=0

(−1)K

J !K!ηJ [η⋆]KIJK = 〈∑∞

J=0[ηB†]J

J!

∑∞K=0

[−η⋆B]K

K! 〉 = 〈eηB†

e−η⋆B〉 . (39)

A two-dimensional Fourier transformation of this produces an alternative method [3] to express the P (β) function:

P (β) =1

π2

∫

d2η χN (η) eη⋆β−ηβ⋆

, P (β) ∈ R ,

∫

d2β P (β) = 1 , (40)

where the two last last steps follow directly from the properties of the Fourier transformation and χN (0) = 1 resultingfrom Eq. (39).

We see that P (β) is a properly normalized quasi-probability distribution satisfying (12). For example, we findP|β0〉(β) = δ(2)(β−β0) when the transformation, Eq. (37), is applied to a coherent state |β0〉. However, as mentioned


9

above, we should keep in mind that coherent states are described by precise distribution only if we restrict ourselvesto normally ordered expectation values. For any other representation, the corresponding coherent-state distributionhas fluctuations. Therefore, even though coherent states always produce the precise classical factorization (32) fornormally-ordered expectation values, the quadrature fluctuations never vanish.

Despite P (β) appears to behave extremely classically, we know that the decomposition (37) is valid for any state.This includes also the nontrivial wave-function superpositions states, such as Schrodinger cat states (discussed inSec. IVC) which exhibit quantum interferences. Other interesting states include those that exhibit squeezing where,e.g., the x quadrature is below the classically allowed ∆xcoh. Since such nonclassical states contain effects beyondclassical averaging, the P (β)-functionmust necessarily become a quasi-probability distribution for true quantum sources

in order to describe the physics beyond classical averaging discussed in Sec. I C.Generally, there are two classes of states depending on whether P (β) is i) strictly positive or ii) partially negative-

valued in some regions of the phase space. Fields with completely positive P (β) are semiclassical because thenEq. (37) indeed yields classical averaging among classical states, which cannot generate new quantum effects beyondthose already described by the coherent states themselves, see Sec. I C. Consequently, the transition from positiveto partially negative-valued P (β) is often considered as the principal definition to identify the boundary betweenquantum and classical states. In fact, P (β) of quantum sources is not only a partially negative but also a nonanalyticfunction. This property makes the numerical evaluation of Eq. (37) challenging.

III. FROM CLASSICAL TO QUANTUM-OPTICAL SPECTROSCOPY

As a generic measurement setup, we consider a situation where the matter system is initially in its ground statebefore it is excited by a strong single-mode laser. The excitation configuration is then probed noninvasively, e.g., byanother weak laser at a predetermined pump-probe delay. This scenario is illustrated schematically in Fig. 3. Sincethe matter is assumed to be initially in its ground state ρmat(0) and

ρ(0) = ρmat(0)⊗ ρlight(0) =

∫

d2β P (β) ρmat(0)⊗ |β〉〈β| ≡∫

d2β P (β) ρ|β〉(0) , (41)

where we expressed the quantum statistics of the pump laser using the decomposition (36). We recognize thatρmat(0) ⊗ |β〉〈β| is actually the total initial state for coherent-state laser excitation ρ|β〉(0). Since the probe isconsidered noninvasive, i.e. it does not appreciably change the system, we do not need to included its quantumstatistics to understand the principal consequences of quantum-optical spectroscopy.

Many-bodysystem

Pump laser

Detection ofproperty R

Probe

Dt

FIG. 3: Schematic representation of laser spectroscopy for a typical pump-probe setup. The many-body system (spheres) isexcited via a strong single-mode pump pulse (white arrow + Gaussian pulse). The resulting many-body excitations are monitored by aweak probe (blue arrow + Gaussian pulse). The probed many-body state induces changes to the probe that are recorded by a detectorthat can, e.g., be a spectrometer, a photodiode, or a streak camera.

The initial state (41) has exactly the form of Eq. (11). Here, the state sum becomes an integral over coherentstates and the appearing weight is a quasiprobability distribution P (α) defined by the Glauber-Sudarshan function.Therefore, we can directly apply the linearity of quantum mechanics. More specifically, Eqs. (14)–(15) and (41)provide the general relation,

RQM ≡ 〈R(t)〉 =∫

d2β P (β)R|β〉 , (42)


10

between the probe response R|β〉 ≡ R|β〉(t) measured using a classical pump source and the probe response RQM toany quantum source having quantum statistics P (β). We omit here the explicit time dependence because the proberecords the outcome during the time interval specified by the detection setup. For example, a specific pump-probedelay ∆t sets the time to be ∆t after the excitation.Since one can routinely measure very different R|β〉 responses in current experimental setups, the relation (42)

provides a rigorous transformation allowing us to access the quantum-optical responses of any system, provided thatthe phase-space integral can be evaluated accurately enough. In our Article, we apply this formulation to realizequantum-optical spectroscopy in semiconductors.

A. Differential spectroscopy

Our Article demonstrates the applicability of quantum-optical spectroscopy in semiconductors by analyzing spectralchanges induced via Schrodinger cat states. More specifically, we analyze the spectral changes caused by a smallperturbation around a classical laser with |β0〉. To model this situation, it is useful to start from Eq. (42) andTaylor expand the appearing classical response R|β〉 around the phase-space point β0. This Taylor expansion is twodimensional and can be evaluated in terms of independent variables β and β⋆. This way, Eq. (42) produces

RQM =

∫

d2β P (β)

∞∑

J,K=0

1

J !K!

[

(

∂∂β⋆

)J (

∂∂β

)K

R|β〉

]

β=β0

(β⋆ − β⋆0)

J(β − β0)

K ≡∞∑

J,K=0

1

J !K!RJ

K δIJK , (43)

where we identified the Taylor-expansion coefficient,

RJK ≡

[

(

∂∂β⋆

)J (

∂∂β

)K

R|β〉

]

β=β0

, (44)

at the displacement center β0 of the laser. The remaining contribution includes the phase-space integral

δIJK ≡∫

d2β (β⋆ − β⋆0)

J(β − β0)

KP (β) = 〈

(

B† − β⋆0

)J(B − β0)

K〉 . (45)

Here, the last step follows by applying the property (38) several times, i.e., we make use of the fact that all phase-spaceintegrals with P (β) produce normally-ordered expectation values.

We find another interesting form for δIJK when we consider the basic properties of the displacement operator

D†(−β0)BD(−β0) = B − β0 , D†(−β0)B†D(−β0) = B† − β⋆0 , D†(−β0)D(−β0) = I . (46)

By applying these in Eq. (45) as well as the cyclic permutation of operators within the trace, we eventually find

δIJK = 〈D†(−β0)(

B†)J BKD(−β0)〉 = Tr[

(

B†)J BK D(−β0)ρlightD†(−β0)]

≡ Tr[

(

B†)J BK ρ|β0〉light

]

≡ 〈(

B†)J BK〉β0. (47)

This becomes a normally ordered expectation value for the source when it is shifted toward the phase-space originby the amount corresponding to the laser’s original displacement β0. For a coherent-state source, this unitary trans-formation D(−β0)|β0〉〈β0|D†(−β0) produces a vacuum state |0〉〈0|. The corresponding δIJK vanishes for all J andK except for J = K = 0 that produces δI00 = 1 for all excitations. In the same way, small quantum-optical laserperturbations yield

ρ|β0〉light ≡ D(−β0)ρlightD†(−β0) , (48)

that is a phase-space distribution in the vicinity of the vacuum state. Clearly, weak perturbations always producesmall δIJK on the scale of the laser’s initial IJK = (β⋆

0)JβK

0 computed using Eq. (16).Since we are interested to study the quantum-optical response for a photon-number N0 = |β0|2 where the system

response changes appreciably, we express Eq. (43) in the form

RQM =

∞∑

J,K=0

RJK N

J+K2

0

J !K!

[

N− J+K

20 δIJK

]

. (49)


11

Here, N− J+K

20 δIJK constitutes a normalized perturbative quantity. For the weak perturbations used in our Article, this

contribution scales proportional to N− J+K

20 because the fluctuations describe only single- or zero-photon perturbations

to the laser. Therefore, the perturbative properties of RQM are obtained by truncating the J and K sums in Eq. (49)after the lowest contributing order in J and K. For example, contributions up to second-order produce

R2ndQM =

∑

J,K=0

RJK

J !K!δIJK

∣

∣

∣

∣

J+K≤2

= R00 +R1

0 δI10 +R0

1 δI01 +R1

1 δI11 +

1

2

(

R20 δI

20 +R0

2 δI02

)

(50)

that is simply truncated after the quadratic terms.In our Article, we project the quantum-optical response from classical measurements whose results only depend on

the photon number of the laser, i.e. R|β〉 = R(N = |β|2). In this situation, the differentiation (44) with respect to βand β⋆ yields

R00 = R(N0) , R0

1 ≡ ∂∂βR(ββ⋆)

∣

∣

∣

β0

= β⋆0

∂∂NR(N)

∣

∣

N0, R1

0 = β0∂

∂NR(N)

∣

∣

N0

R11 = ∂

∂NR(N)

∣

∣

N0+N0

∂2

∂N2R(N)∣

∣

∣

N0

, R02 = (β⋆

0)2 ∂2

∂N2R(N)∣

∣

∣

N0

R20 = β2

0∂2

∂N2R(N)∣

∣

∣

N0

. (51)

By inserting this into Eq. (50), we obtain

R2ndQM = R(N0) +

∂R(N)

∂N

∣

∣

∣

∣

N0

(

β⋆0δI

01 + β0δI

10 + δI11

)

+1

2

∂2R(N)

∂N2

∣

∣

∣

∣

N0

(

2N0δI11 + (β⋆

0)2δI02 + β2

0δI20

)

, (52)

after we properly reorganize the contributions multiplying each ∂JR(N)∂NJ .

The quantum-optical laser perturbation defines the different δIJK contributions. Their explicit form follows fromEq. (45), producing

δI01 = 〈B − β0〉 = 〈B〉 − β0 , δI10 = (δI01 )⋆ = 〈B†〉 − β⋆

0 ,

δI11 = 〈(B† − β⋆0)(B − β0)〉 = 〈B†B〉 − β0〈B†〉 − β⋆

0〈B〉+ |β0|2 ,δI02 = 〈(B − β0)

2〉 = 〈BB〉 − 2β0〈B〉+ β20 , δI20 = (δI02 )

⋆ = 〈B†B†〉 − 2β⋆0〈B†〉+ (β⋆

0)2 . (53)

With theses results, we convert the first- and second-order contribution of Eq. (52) into

1st ≡ β⋆0δI

01 + β0δI

10 + δI11 = 〈B†B〉 − |β0|2 = N −N0 ,

2nd ≡ 2N0δI11 + (β⋆

0)2δI02 + β2

0δI20

= 2N0

(

〈B†B〉 − β0〈B†〉 − β⋆〈B〉+ |β0|2)

+ 2Re[

(β⋆0)

2(

〈BB〉 − 2β0〈B〉+ β20

)]

, (54)

where the first-order contribution is simply the photon number difference between the perturbed (N = 〈B†B〉) andthe initial (N0 = |β0|2) laser. The second contribution can be simplified further when we apply Eq. (29) to express〈B†B〉 = ∆〈B†B〉 + 〈B†〉〈B〉 and 〈BB〉 = ∆〈BB〉 + 〈B〉〈B〉 in terms of its correlation and classical factorizationparts. This step leads to

2nd = 2Re[

N0|〈B〉 − β0|2 + (β⋆0)

2(〈B〉 − β0)2]

+ 2Re[

N0∆〈B†B〉+ (β⋆0)

2∆〈BB〉]

(55)

which is always a real-valued quantity.At this point, it is beneficial to express the photon operators via a unitary transformation, B = B‖eiφ0 where

φ0 defines the phase direction of the unperturbed laser, i.e. β0 = |β0|eiφ0 . With this choice, B‖ = x‖ + iy‖ is aBoson operator whose x‖ quadrature is aligned with β0 while y‖ is perpendicular to it. By implementing this unitarytransformation into Eq. (55), we obtain

2nd = 2N0Re[

∣

∣〈B‖〉 − |β0|∣

∣

2+

(

〈B‖〉 − |β0|)2]

+ 2N0Re[

∆〈B†‖B‖〉+∆〈B‖B‖〉

]

. (56)

The perturbed displacement 〈B‖〉 ≡ β‖ + iβ⊥ can be expressed in terms of components that are parallel (β‖)and perpendicular (β⊥) to the β0 displacement. We also recognize that the correlated part of Eq. (56) yields

Re[

∆〈B†‖B‖〉+∆〈B‖B‖〉

]

= 2(

∆x2‖ −∆x2coh

)

, based on Eq. (29). In this expression, ∆x‖ and ∆xcoh define the

quadrature fluctuation of the perturbed and unperturbed laser, respectively, in the direction of β0. With theseconsiderations, Eq. (56) reduces to

2nd = 4N0

(

(

β‖ − |β0|)2

+∆x2‖ −∆x2coh

)

, (57)


12

after we evaluate the real part explicitly.As results (54) and (57) are inserted into Eq. (52), we obtain the quantum-optical response

R2ndQM = R(N0) +

∂R(N)

∂N

∣

∣

∣

∣

N0

(N −N0) +1

2!

∂2R(N)

∂N2

∣

∣

∣

∣

N0

4N0

[

(β‖ − |β0|)2 +∆x2‖ −∆x2coh

]

, (58)

up to the perturbations in second order. We see that the second term in Eq. (58) appears due to the quantization oflight because it is sensitive to the quadrature fluctuations of the laser. This introduces new quantum-optical aspectsto the spectroscopy, as discussed in our Article. Since the quantum corrections to the differential spectroscopy stemform the quadratic contribution, it is clear that only a nonlinear R(N) can produce quantum-optical effects. Equation(58) defines a general expression for small quantum-optical perturbations made around any classical measurementwhose response depends only on intensity.

In a case where the nonlinear classical response also depends on the phase, one must start the perturbation analysisfrom the general Eq. (50) to explain the emerging nonclassical features. To determine the corresponding R2nd

QM, we

apply the factorization (29) to Eq. (53), yielding

δI01 = 〈B〉 − β0 ≡ ∆β , δI10 = ∆β⋆ ,

δI11 = ∆〈B†B〉+ |∆β|2 , δI02 = ∆〈BB〉+∆β2 , δI20 = ∆〈B†B†〉+ (∆β⋆)2, (59)

where ∆β defines the perturbation in the laser’s displacement. By inserting Eq. (59) into Eq. (50), we get

R2ndQM = R0

0 +R10 ∆β

⋆ +R01 ∆β +R1

1 |∆β|2 +R2

0 (∆β⋆)2+R0

2 ∆β2

2+R1

1∆〈B†B〉+ R20 ∆〈B†B†〉+R0

2 ∆〈BB〉2

=∑

J,K=0

RJK

J !K!(∆β⋆)

J∆βK

∣

∣

∣

∣

J+K≤2

+Re[

R11∆〈B†B〉+R0

2 ∆〈BB〉]

= R2nd|β0+∆β〉 +Re

[

R11∆〈B†B〉+R0

2 ∆〈BB〉]

(60)

because the ∆β terms construct the Taylor expansion of R|β〉 around β0. Therefore, we may identify R2nd|β0+∆β〉

as the classical response R|β0+∆β〉 evaluated up to second order in the ∆β perturbation. The remaining nonlinearcontributions are nonclassical because they depend on the quantum statistics of the perturbed source.

To define the physical form of the quantum-optical response within R2ndQM, we implement a unitary transformation

B = B2 eiφ2 where the phase direction is defined from the direction of the R0

2 = |R02|e−2iφ2 contribution. As this is

applied together with the identification of the quadrature fluctuations (29), we find

∆〈B†B〉 = ∆〈B†2B2〉 = ∆x22 −∆x2coh +∆y22 −∆x2coh ,

Re[

R02 ∆〈BB〉

]

= |R02|Re [∆〈B2B2〉] = |R0

2|(

∆x22 −∆y22)

. (61)

This result converts the general response (60) into

R2ndQM = R2nd

|β0+∆β〉 +R11

(

∆x22 −∆x2coh +∆y22 −∆x2coh)

+ |R02|(

∆x22 −∆y22)

(62)

that is defined up to the second order in the laser perturbations. Since the nonlinear R11 and R0

2 are not necessarilyconnected, the quantum-optical perturbations to phase-sensitive measurements produce a quantum-optical responsethat depends on both the ∆x2 and ∆y2 fluctuations of the quantum source. As before, R2nd

|β0+∆β〉 represents the classicalresponse while the remaining quantum-optical contributions stem exclusively from the nonlinearity of the response.

The discussion above can be directly generalized to included higher-order corrections to the perturbation expansion.However, we do not present this analysis here mainly because we have applied the perturbative approach only as aninterpretation tool in our Article. In other words, all responses presented in our Article are obtained using the cluster-expansion transformation in a form that is valid to all orders of the quantum-optical perturbations, as discussed next.

B. Quantum-optical responses using the cluster-expansion transformation

Even though the transformation (42) in principle allows us to construct the quantum-optical response from acomplete set of classical measurements, we are faced with some serious challenges before we can really access RQM


13

from actual measurements. Formally, Eq. (42) contains an integral throughout the entire phase space. However, oneobviously cannot measure the classical response R|β〉 continuously for all β values. Furthermore, any measurement isbound to have some noise and any quantum source has a nonanalytic Glauber-Sudarshan function, which complicatesthe evaluation of the convolution integral (42) considerably.

To find a pragmatic solution, we need an intelligent and robust scheme that reliably projects the vast amount ofinformation collected in a set of classical measurements into actual quantum responses RQM. Fortunately, R|β〉 is awell-behaving, real-valued function simply because it results from a measurement. Even though β is the displacementof the laser in a classical measurement, it also assigns a phase-space coordinate of a real-valued R|β〉. Therefore, theclassical response R|β〉 can also be viewed as an unnormalized phase space distribution. In other words, we are allowedto associate the measured response with a Wigner-function like response distribution

WR(β = x+ iy) ≡WR(x, y) ≡R|β=x+iy〉

NR

, NR =

∫

Ω

d2β R|β〉 , (63)

where NR takes care of the normalization and Ω determines the phase-space region where the distribution is defined.Clearly, Ω has to be chosen such that it covers the physically relevant phase space and produces a nonvanishing NR.Fortunately, as we will show below, these requirements can be well satisfied once the relevant R|β〉 data have beencollected.

As the central tool, we use the highly accurate and efficient cluster-expansion transformation (CET)[24] thatconverts phase-space distributions defined within a finite Ω and a discrete number of points into analytic phase-space distributions. In the following, we outline the pragmatic steps needed to implement the CET for the specificsituation of classical measurements whose outcome is controlled via the pump laser’s photon number. In this situation,WR(x, y) =WR(|β|) is rotationally symmetric which implies vanishing first-order moments

∫

dx dy xWR(|x+ iy|) =∫

dx dy yWR(|x+ iy|) = 0. (64)

The second-order moments

∫

dx dy x2WR(|x+ iy|) =∫

dx dy y2WR(|x+ iy|) = ∆x2R, (65)

are identical in the x and y direction. They identify the variance ∆xR of the distribution.We concentrate here on two physically relevant situations where WR(|x+ iy|) either approaches a constant level or

decays as β is increased. The constant level implies complete linearity of the response in this intensity range. Basedon Eq. (58), we find no quantum-optical sensitivity in this intensity regime. Therefore, we can remove this trivialconstant part from R|β〉. After this step is included in (63), WR(|x+ iy|) decays for large enough displacements. Forphysically relevant systems, the decay is fast for |β| ≫ ∆xR. Therefore, one needs to measure R|β〉 only over an areathat covers a radius of a few ∆xR. In other words, knowing ∆xR defines a pragmatic way how to perform classicalmeasurements that cover a sufficiently broad Ω area in the phase space.

After these considerations, one can proceed to determine also the higher-order moments of the WR(|β|) distribution

∫

Ω

d2β |β|2J WR(|β|) = I(J) . (66)

These integrals can be evaluated by using a straightforward discretization of the phase space and by limiting the extentof the phase-space to Ω. After the I(J) values are known, they uniquely define the coefficients for the correlations

aR(J) =J !J !

[4∆x2R]J

J∑

L=0

[−2∆x2R]LI(J − L)

L!(J − L)!(J − L)!. (67)

The explicit derivation of this transformation is presented in Ref. [24].This scheme converts the measured distribution into aR(J) coefficients that contain correlations, i.e. cumulants

such as kurtosis and skewness of the distribution up to rank J which uniquely characterizes any distribution. Thehigher-order correlations are either irrelevant or describe only the experimental noise. We can, therefore, use thecluster number C to truncate the number of relevant correlations in the description. The last step of CET then


14

introduces a cluster-expansion truncation [24] to express the measured distribution up to C clusters:

RCET(β) =NR

4∆x2R

C2

∑

J=0

(−1)JaR(J)WJ

(

|β|24∆x2

R

)

, WJ (x) =2

πe−2x

J∑

k=0

22J−k(−x)J−k

k!(J − k)!(J − k)!. (68)

As a simple condition, C should be chosen high enough such that RCET(β) follows accurately the measured R|β〉.

Most importantly, RCET(β) contains a Gaussian exp[− |β|22∆x2

R

] factor multiplied by a polynomial determined by the

correlations. Due to the Gaussian factor, Eq. (42) can be analytically integrated even when P (α) corresponds to any

quantum source. As the only requirement, we find that the measurement must satisfy the condition

∆xR ≥ 12 = ∆xcoh ⇔

∫

d2β P (β)RCET(β) = RQM is analytic ∀ physical P (β) . (69)

This means that the width of the classical measurement outcome R|β〉 cannot be lower than the minimum level of theclassical quadrature fluctuations ∆xcoh. Loosely speaking, ∆xcoh also defies the width of the ”classical” single-photonfluctuations in the phase space. Therefore, the large-scale changes of R|β〉 cannot happen faster on a single-photonscale. This restriction is very natural because even the most nonlinear system must absorb at least one photon beforetotal saturation is reached.

The CET approach can also handle phase sensitive measurements. The association of a measured R|β〉 with aresponse distribution excatly follows from the relation (63). The actual extent of the measured β sweeps can bedetermined by making sure that WR(β) covers a sufficiently large phase-space region around its center. As discussedbelow Eq. (65), the measured quadrature fluctuations ∆xR and ∆yR of WR(β) give a good estimate about the phasespace Ω that the experiments should cover. Once WR(β) is measured, one can determine the normally-orderedexpectation values

IJK =

∫

Ω

d2β [β⋆]JβK WR(β) . (70)

In analogy to the evaluation of (66), we can use a straightforward discretization of the phase space with the onlymodification that the integrals are now two-dimensional and, hence, more extensive in terms of computation time.The found IJK can then be converted into cluster-expansion coefficients aJK using the most general CET form presentedin Ref. [24]. Due to the higher dimensionality of the problem, aJK becomes a two-index tensor in contrast to the purelyintensity-depend problems that can be solved using a one-dimensional aR(J).

Once the coefficients aJK are known, the quantum-optical response follows from a formula analogous to Eq. (68).Again, the one-dimensional sum has to be replaced by a two-dimensional one and instead of WJ , we have a two-indexW J

K(β) function presented in Ref. [24]. Especially, W JK(β) has an analogous structure as WJ (x) with a polynomial

multiplying an Gaussian e−|β|2 . Therefore, the computation of the quantum-optical response with Eq. (69) benefitsfrom the same intrinsic convergence as the analysis made for the intensity-dependent measurements. In other words,the presented CET approach (63)–(69) can be directly generalized to project any sufficiently complete set of classicalmeasurements into quantum-optical responses. From the technical point of view, the only additional effort lies in themeasurements and the computations because both must be executed on a two-dimensional grid.

C. CET reconstruction of quantum-Rabi fopping

As a tutorial illustration of our CET scheme and in order to benchmark the CET projection for a standardquantum-optical example, we analyze how it works with the Jaynes-Cumming model [25]

H = ~ωB†B + ~ω21σz − g(

σ−B† + σ+B

)

, (71)

that describes an atomic two-level system (TLS) coupled to a single quantized light mode. The appearing σj are

given by the usual Pauli spin matrices and g defines the strength of light-matter coupling. The interaction part of Hdescribes the process where a single-photon absorption excites the atom from the ground to the excited state (B σ+)and its reverse B† σ−. Such a strong-coupling scenario has been realized experimentally, e.g., by positioning one atominside a high-quality cavity [26, 27] that couples with a single atomic transition at ~ω21.We assume that the TLS is initially in its ground state before it is excited resonantly (ω = ω21) with a coherent

state |β〉. In this case, the initial state is ρ(0) = ρlight⊗|−〉〈−| where |−〉 is the ground state of the TLS. As we probe


15

the population inversion Rz(t) ≡ 〈σz〉 of the atomic system at a given time t, we record the response R|β〉 ≡ Rz(t, |β〉)as function of the classical laser excitation |β〉. Repeating these measurements for many excitation levels constructsthe complete set of classical input data for the CET projection of the quantum response.

The Hamiltonian (71) produces a relatively simple quantum dynamics [3] for the population inversion

Rz(t, |β〉) =∞∑

n=1

pn sin2gnt−1

2, gn ≡ g

√n , (72)

where pn is the photon statistics of the light source used. Due to the light quantization, Rz(t, |β〉) contains onlydiscretized frequencies gn resulting from anharmonic quantum-Rabi flopping. Any coherent-state source |β〉 has aPoissonian photon statistics [3]

pcohn =|β|2nn!

e−|β|2 , (73)

such that Rz(t, |β〉) depends only on the average photon number N = |β|2. We insert this into Eq. (72) and solveRz(t, |β〉) by summing numerically over 240 |β| values within the interval β ∈ [0, 10]. The resulting Rz is shown inFig. 4A as function of time and β.

TLS

invers

ion

Rz

Time Time

Time Time

0

original + noise

0 02 24 46 6

0 02 2

4 4

0 0

+1/2

-1/2

+1/2

-1/2

1/2

-1/2

0

0

11

22

Am

plit

ude

Input CETa

c d

b

FIG. 4: Jaynes-Cummings Model described in the CET framework. a, Classical source: The time evolution of Rz(t, |β〉) isshown as function of β. The red (blue) line corresponds to the excitation with |β|2 = 2 (5) photons on average. b, Quantum source: TheCET-projected time evolution of the population inversion is plotted for the Fock states |1〉 to |5〉, solid lines from top to bottom. Theintermediate values correspond to states (1− p)|n〉〈n|+ p|n+1〉〈n+1| with n = n+ p photons on average, p ∈ [0, 1]. The field amplitudeis

√n or |β|. c, The dynamics of Rz(t, |β〉) after the excitation with a classical |β =

√2〉 source is shown as shaded area. The black line

represents the corresponding input data with ±0.08 noise added. d, Comparison between the quantum response to the Fock-state |n = 2〉CET projected from the original data (shaded area) and the result where ±0.08 (black line) noise has been added. The input data iscalculated using 240 β values within 0 < |β| < 9 for each t. The largest cluster number is C = 80.

The temporal evolution of Rz(t, |β〉) produces the well-known collapse and revival of the oscillations [28, 29] before itbecomes nearly chaotic. From the CET point of view, we also see that the β dependency of Rz(t, |β〉) produces a ∆xRextension that is larger than 1

2 . Therefore, the condition (69) is well satisfied and produces converging CET integrals.

Only at a few discrete times, ∆xR approaches ∆xcoh = 12 , which indication of the fact that the TLS occasionally

exhibits extreme quantum-optical nonlinearities. Since even a single photon can yield major nonlinear changes in aTLS it is not surprising that many famous quantum-optical demonstration experiments have been performed usingmatter that essentially behaves like a TLS.

Since Fock-states |n〉 have a nonanalytic and partially negative-valued P|n〉(β), we choose them as quantum sources.We do not solve RQM ≡ Rz(t, |n〉) analytically — which would be trivial — but CET project it from the discretizedRz(t, |β〉) data. In other words, we convert the recorded set of Rz(t, |β〉) into aR(J) coefficients using Eqs. (65)–(67).We also have integrated Eq. (69) with the Glauber-Sudarshan function of a given Fock state yielding


16

R|n〉CET(t) =

NR

4∆x2R

C2

∑

J=0

(−1)JaR(J)FJ , FJ ≡n∑

k=0

n!(J + k)!

k!k!J !J !(n− J)!

( −1

2∆x2R

)k

. (74)

where both ∆xR and aR(J) at a fixed time are constructed from the discrete set of Rz(t, |β〉) data.Figure 4b shows the CET projected R

|n〉CET(t) (solid lines) up to |n = 5〉. The intermediate surfaces correspond to

states ρlight = (1−x)|n〉〈n|+x|n+ 1〉〈n+ 1| with the average photon number n+x. This exercise shows that, despitethe finite number of |β〉 states included, the CET projection accurately reconstructs the well-known quantum Rabiflopping [26],

R|n〉z (t) = sin2

√n g t− 1

2 , (75)

i.e. sinusoidal oscillations whose frequency increases proportional to√n. This result follows also trivially from Eq. (72)

because the Fock state’s photon statistics is p|n〉n′ = δn,n′ .

Altogether, this little exercise demonstrates that our CET approach can indeed project nontrivial quantum responsesfrom discrete outcomes R|β〉 of classical spectroscopy. To test the robustness of the scheme, we intentionally addrandom noise to the classical-source input data to mimic experiments. Figure 4c presents exemplary Rz(t, |β〉) slicesat |β|2 = 2 with (line) and without the added noise (shaded area). We CET project the full noisy data set R(t, |β〉)to construct the corresponding Fock-state |n = 2〉 response shown in Fig. 4D. As we can see, the CET projectionaccurately reconstructs the correct form of the quantum Rabi flopping even from the noisy data.

D. Atomic vs. quantum-well absorption spectroscopy

To relate the TLS model results to quantum-well (QW) absorption measurements, we consider probe spectra thatcan, in principle, be measured using a configuration like that illustrated in Fig. 3. Based of Ref. [30], the probeabsorption of an atomic two-level system is given by

αTLS(ω) ∝ −Im

[

Rz

~ω21 − ~ω − iγ

]

, (76)

where the dephasing γ does not depend on the excitation if it stems dominantly from the atom’s coupling to areservoir. This is realized as long as the measurement isolates a single atom which is at most weakly coupled to otheratoms. Confining this atom within a cavity while it is probed, realizes a system whose excitation dynamics can bedescribed within the Jaynes-Cummings model (71).

In this situation, the classical pump generates a state inversion that depends strongly on the pump excitation aswell as on the pump-probe delay. The full Rz = Rz(∆t, |β〉) is shown in Fig. 4. The measured absorption establishesone possibility to deduce the dynamics of the inversion factor as function of time and pump intensity in the realm ofclassical laser spectroscopy. At the same time, we see that all absorption nonlinearities exclusively stem from the Rz

response of the TLS. Already a single-photon modification of the laser may induce a large change of the TLS response.A standard pump-probe measurement on semiconductor QWs determines a QW absorption [31]

αQW(ω) ∝ Im

[

∑

λ

osc[ρMB]

Eλ[ρMB]− ~ω − iγλ[ρMB]

]

, (77)

where the oscillator strength osc[ρMB], absorption resonances Eλ[ρMB], and dephasing function γλ[ρMB] are compli-cated functionals of the many-body state ρMB excited by the pump laser.[32] The nonlinearities of osc[ρMB], Eλ[ρMB],and γλ[ρMB] emerge only when the classical laser excitation creates a substantial number of electron-hole pairs in theQW. Therefore, a single-photon change of the laser intensity can only lead to weak modifications in semiconductorQWs. Therefore, the identification of quantum-optical effects in QWs can be considered as a challenging task.

However, it helps to realize that the quantum-optical signatures of the QW response are contained in all quantities,osc[ρMB], Eλ[ρMB], and γλ[ρMB] defining the QW absorption, not only in the oscillator strength like for a TLS.Therefore, quantum-optical αQW(ω) spectroscopy can access many-body phenomena not present in single atoms. Inother words, TLS and many-body quantum optics investigate quite different aspects of quantum phenomena inducedvia the light-matter interaction. Obviously, quantum optics is more complicated in QWs than in TLS because thematter states involved are more complicated. As a reward, however, semiconductor quantum optics can access abroader range of phenomena than TLS, simply because they are strongly interacting many-body systems. We haveshow in our Article that one can uniquely use quantum-optical signatures to characterize and control the specificmany-body correlations in semiconductor QWs.


17

IV. PROJECTED QUANTUM-OPTICAL RESPONSES TO SEMICLASSICAL AND QUANTUM

SOURCES

Before we demonstrate the experimental applicability of the CET method, we first summarize some the mainfeatures of quantum and semiclassical sources. We start with Gaussian which can be viewed as a generalization ofcoherent states,

WGauss(x, y) ≡ 1

2π∆x∆ye−

(x−β1)2

2∆x2 e− (y−β2)2

2∆y2 . (78)

The Gaussian states are uniquely defined by the displacement β = β1 + iβ2 and the quadrature fluctuations ∆x and∆y and we used the freedom to rotate the phase-space direction such that the main axes of the Gaussian fluctuationsare aligned with the x and y directions. Alternatively, one can parametrize the fluctuations of the Gaussian state viaphoton number and two-photon absorption correlations

∆〈B†B〉 = ∆x2 +∆y2 − 12 , Re [∆〈BB〉] = ∆x2 −∆y2 , (79)

respectively, after one inverts Eq. (29). Since we have chosen x and y to be the main axes of the quadrature fluctuation,∆〈BB〉 must be a real-valued quantity in the phase-space direction chosen.Regardless of how we parametrizeWGauss(x, y), the quadrature fluctuations must satisfy the Heisenberg uncertainty

relation (28) ∆x∆y ≥ 14 . For the equal sign, the WGauss(x, y) are minimum uncertainty states. In particular,

coherent states are the minimum uncertainty states with equal fluctuations in all directions. This class also includesWGauss(x, y) whose ∆x is squeezed below the classically allowed ∆xcoh when the complementary ∆y fluctuation arestretched. Such minimum uncertainty states are commonly referred to as squeezed states.

A. Thermal source

A thermal light source has a WGauss(x, y) that is parametrized completely by the photon-number correlations.The average photon number is N = 〈B†B〉 = ∆〈B†B〉 while both β and ∆〈BB〉 vanish. Substituting this intoEqs. (78)–(79), the Wigner function becomes

Wth(x, y) ≡ 2π(2N+1) e

− 2(x2+y2)2N+1 . (80)

This state has exceedingly large fluctuations in phase space. In fact, the thermal state is completely defined by itsfluctuations because its classical amplitude vanishes.

The thermal state’s normally ordered characteristic function is χthN (η) = e−N |η|2 [24]. Inserting this into Eq. (40)

produces the Glauber-Sudarshan function

Pth(x, y) ≡ 1πN

e−(x2+y2)

N (81)

which is positive definite and well behaved. A thermal state can be understood to describe a semiclassical state becauseits properties can be explained through classical averaging of coherent states as discussed in Sec. II C. According toEq. (69), the convolution of Pth(x, y) and CET Eq. (68) produces the response

Rth =NR

π(N + 2∆x2R)

C2

∑

J=0

aR(J)

J !

( −1

N + 2∆x2R

)J

. (82)

to a thermal source that can be projected from a set of classical measurements. The contributions for J > 1 areclearly different from a classical response (68), indicating that nonlinear contributions can indeed be sensitive to thequantum statistics.

B. Squeezed vacuum

It is also interesting to consider Gaussian sources which have strong squeezing but no classical displacement becauseone can then strongly enhance the quantum fluctuations. For this purpose, we set β1 = β2 = 0 in Eq. (78) and use


18

the maximum value ∆〈BB〉 =√N2 +N allowed by Eqs. (28)–(29) for a given ∆〈B†B〉 = N . These choices produce

the so-called squeezed vacuum

Wsqz(x, y) ≡ 1

2πe−

x2

2∆x2 e− y2

2∆y2 , ∆x2 = 14 + N+

√N2+N2 , ∆y2 = 1

4 + N−√N2+N2 , (83)

that is a minimal-uncertainty state. Interestingly, a squeezed vacuum may contain arbitrarily many photons onaverage. Not surprisingly, the properties of such a state deviate strongly from an actual vacuum (N = 0) once(N ≫ 1). For example, Eq. (83) produces ∆y2 → 1

16N for large N while ∆x2 → N . For a high-intensity squeezed-

vacuum source with N = 106 photons, the ∆y is 2000 times narrower than that of a coherent state. Therefore, ahigh-intensity squeezed vacuum approaches nearly perfect accuracy in the y direction while the x accuracy is verylow.

The squeezed vacuum has a normally-ordered characteristic function χsqzN (η) = e−2[∆x2− 1

4 ]η22e−2[∆y2− 1

4 ]η21 .[24] We

notice that this has a Gaussian form and [∆y2 − 14 ] is strongly negative valued for squeezing. This means that the

part e−2[∆y2− 14 ]η

21 of χsqz

N (η) increases without bound for large values of η1. Therefore, the Fourier transformation,i.e. Eq. (40), produces an extremely divergent Glauber-Sudarshan function

Psqz(x, y) =√

2π(2∆x2−1) e

− 2β21

4∆x2−12

π

∫ ∞

0

dη1 e+[ 12−2∆y2] η2

1 cos 2iβ2η1 . (84)

The resulting Psqz(x, y) diverges both to positive and negative infinity because the integration argument does thattoo. Therefore, the squeezed vacuum must necessarily be a quantum source as discussed in Sec. II C.Even though Psqz(x, y) is divergent, it produces a completely converging convolution integral (68) such that we can

project measurements to the quantum response,

Rsqz =NR

2π√S2 − s2

C2

∑

J=0

aR(J)CJ , CJ =1

[4(S2 − s2)]J

[ J2 ]∑

k=0

[2S]J−2ks2k

k!k!(J − 2k)!,

S ≡ ∆x2R + N2 , s ≡

√N2+N2 , (85)

for any squeezed-vacuum state excitation. The identified s defines the level of squeezing while S combines the quantumstatistical width of the classically recorded response and the intensity fluctuations N of the source. Only if we sets = 0 independent of S, Eq. (85) reproduces the result (82) for the thermal source. Therefore, the squeezed sourceproduces a nonlinear response that is distinctly different from that of any classical or semiclassical source as shownin the main article.

C. Schrodinger cat states

In our Article, we use Schrodinger cat states [3] to demonstrate the transition from classical to quantum-opticalspectroscopy. These cat states are the superposition of two different coherent states

|β, γ, θ〉 = N(

e−iIm[βγ⋆]|Γ−〉+ eiθ eiIm[βγ⋆]|Γ+〉)

, N ≡ 1√2 + 2 e−2|γ|2 cos θ

, Γ± ≡ β ± γ . (86)

where Γ± defines which coherent states appear in the superposition. Here, we have slightly generalized the catstates used in the Article by allowing for an arbitrary phase θ to appear between the coherent-state components.We find the usual squeezing and stretching cat by setting θ = 0 and inserting |β0, i|γ| β0

|β0| , 0〉 and |β0, |γ| β0

|β0| , 0〉,respectively. Physically, the squeezing (stretching) cat squeezes (stretches) the state in the β direction below (above)the minimum-uncertainty limit ∆xcoh = 1

2 .Using the property (32) and the projection (33) multiple times, we eventually find that the state (86) produces the

normally-ordered expectation values

IJK ≡ 〈[B†]JBK〉 = N 2(

(Γ⋆−)

J ΓK− + (Γ⋆

+)J ΓK

+ + e−2|γ|2 [eiθ(Γ⋆−)

J ΓK+ + e−iθ(Γ⋆

+)J (Γ−)

K]

)

. (87)


19

As this is inserted into Eq. (39), we obtain the normally ordered characteristic function

χ|β, γ, θ〉N (η) = N 2

(

eΓ⋆−η−Γ−η⋆

+ eΓ⋆+η−Γ+η⋆

+ e−2|γ|2[

eiθeΓ⋆−η−Γ+η⋆

+ e−iθeΓ⋆+η−Γ−η⋆

])

. (88)

Substituting this into Eqs. (40) produces the Glauber-Sudarshan function,

P |β, γ, θ〉(β′) = N 2[

δ(2)(β′ − Γ−) + δ(2)(β′ − Γ+)]

+ 2N 2 limǫ→0

e−|β′−β|2

ǫ

πǫe(

1ǫ−2)|γ|2 cos

(

θ − 2Im[

(β′−β)γ⋆

ǫ

])

. (89)

Here, the last term is clearly nonanalytic and potentially negative. By making ǫ smaller, P |β, γ, θ〉 diverges towards

negative infinity at the phase-space point β′ = β − i(π−θ)ǫ

γ. Therefore, the state |β, γ, θ〉 must be a genuine quantum

source based on the discussion in Sec. IID.The quantum statistics of |β, γ, θ〉 can alternatively be presented via the Wigner function

W |β, γ, θ〉(β′) =2N 2

π

[

e−2|β′−Γ−|2 + e−2|β′−Γ+|2 + 2 e−2|β′−β|2cos (θ − 4Im [(β′ − β)γ⋆])]

, (90)

obtained by substituting Eq. (89) into Eq. (34) and by evaluating the appearing integral. Figure 5 shows contourplots of the Wigner functions for cat states ||β|, i|γ|, θ〉 as function of six different θ when the fluctuations produce asingle-photon increment (SPI) vs. a two-photon increment (2PI) with respect to a classical laser |β〉. We have chosenhere γ = i|γ| and put the cat state’s displacement center exactly to the origin. The explicit scheme to choose thecorrect β and |γ| values is presented in Sec. IVD.

(i) =0q (i) =0q

(ii) = /3q p (ii)

= /3q p

(iv) q p= (iv)

q p=

(iii) q p=2 /3 (iii)

q p=2 /3

(v) q p=4 /3 (v)

q p=4 /3

(vi) q p=5 /3 (vi)

q p=5 /3

2

2

2

2 22 2

0

0

0

0 00 0

-2

-2

-2-2 -2-2 -2

x quadrature x quadrature

yquadra

ture

a b0.5

0.0

-0.5

FIG. 5: Contour plots of Wigner functions for a selection of Scrodinger cat states. These placement center is set at the originand the γ parameter is chosen along the y axis. We use Eqs. (92)–(93) to uniquely define the β and |γ| parameters in |β, i|γ|, θ〉 for, a, asingle-photon increment (SPI, δN = 1) and, b, a two-photon increment (2PI, δN = 2) states for six representative θ values.

As a general feature, the Schrodinger cat state (86) produces positive lobes at the Γ± displacement centers. Theseare often referred to as the ”ears” of the cat state. Naturally, the ears are further separated as we go from the SPI-to the 2PI-cat state because the photon increment increases monotonously with |γ|, see Sec. IVD. At the same time,the Wigner function produces an interference pattern between the ears where W (x, y) dips to negative values (bluecolors). The appearing interference pattern oscillates faster as the ears of the cat are separated further by increasingthe γ value. This quantum interference pattern is often referred to as the whiskers of the cat state. We see that theexact position and shape of the whiskers is controlled by the θ parameter.

Interestingly, the θ = π cat produces a completely rotationally symmetric distribution for the SPI perturbation inFig. 5a(iv). Its W (x, y) dips down to − 2

π≈ 0.637 while a positive ”donut” ring surrounds this negative region. These

features are the typical hallmarks of a Fock-state |1〉. It is indeed straightforward to show that the limit,

limγ→0

|β0, γ, π〉 = D(β0)|1〉 , (91)


20

produces a displaced Fock state |1〉. Obviously, such a state establishes one possibility to introduce quantum SPI tothe initial |β0〉 source because the corresponding W (x, y) is negative.For the other cases shown in Fig. 5a, the cat states are squeezed in the x direction. As we analyze in Sec. IVD,

the ∆x fluctuations become smaller than classically allowed ∆xcoh = 12 when we use the SPI cat states. Such SPI

perturbations are referred to as squeezing cats in the Article. We can rotate these cats states by π2 in phase space if

we choose γ = |γ|. This situation produces the stretching cat with enhanced quadrature fluctuation, i.e. ∆x > ∆xcoh,in the direction of the laser.

D. Constructing N-photon increment perturbations for the laser

In general, we are interested in a situation where the laser initially has the displacement β0 and the photon-number N0 = |β0|2 on average. To demonstrate quantum-optical spectroscopy in our Article, we then analyze smallperturbations to the laser in the form of Schrodinger cat states. In this context, one can increase the laser’s photonnumber exactly by δN photons using several (β, γ, θ) combinations. To implement such quantum-optical incrementswith states whose classical aspects are identical to those of the initial coherent-state laser |β0〉, we leave the averagedisplacement unchanged, i.e. 〈B〉 = β0. As we apply this condition together with Eq. (87), we find that the cat statemust be constructed via

Γ± = β0 − i δf γ ± γ , δf =sinθ e−2|γ|2

1 + cosθ e−2|γ|2 , (92)

where δf defines how strongly the displacement due to the whiskers must be compensated. Figure 5a shows that theSPI cat generates a minor tilt to the ears whenever θ 6= 0 and θ 6= π due to the additional δf displacement. Thesedistortions are negligible for the 2PI cat (Fig. 5b).

To settle the actual magnitude of the γ parameter, we compute the average photon number 〈B†B〉 = I11 by insertingthe result (92) into Eq. (87),

〈B†B〉cat = |β0|2 + δN , δN = |γ|2 1− e−4|γ|2

[1 + cosθ e−2|γ|2 ]2. (93)

To find the |γ| of the SPI cats, we set δN = 1 in Eq. (93) and determine its roots numerically. Figure 6a(i) presentsδN as function of |γ|2 for θ = 0 (black line), θ = π

2 (red line), and θ = π (blue line). All of them create very differentcurves such that their SPI roots (crossings with the lower horizontal line) are different. The Wigner functions in Fig. 5are obtained using the γ roots of Eq. (93) to determine the desired δN = 1. As a special case, we notice that θ = πproduces a δN increment that remains larger or equal than one while the other θ directions approach δN = 0 forsmall |γ|. This is another way to see the emergence of the displaced Fock state discussed in connection with Eq. (91).

Based on Eq. (58), also the 〈BB〉 of a cat state plays an important role for the quantum-optical response. Thecat-state parameters (92) produce

〈BB〉cat = β20 + γ2

(

1 + δf2)

(94)

as they are inserted into Eq. (87) and the identifications (92) and (93) are used. Since 〈B〉 = β0 for this cat state, wefind that the cat state produces the correlations

∆〈B†B〉cat = δN , ∆〈BB〉cat = γ2(

1 + δf2)

(95)

based on Eqs. (29), (93), and (94). We see that the correlations of the perturbed source completely depend on thecat-state parameters, i.e. the δN increment, the cat-displacement γ, and the angle-parameter θ. Inserting Eq. (95)into (29), we determine the quadrature fluctuations of the cat state

∆x2 = ∆x2coh + 12Re

[

δN + γ2(

1 + δf2)]

∆y2 = ∆x2coh + 12Re

[

δN − γ2(

1 + δf2)]

. (96)

Without loss of generality, we can choose the direction of the initial laser β0 along the x axis such that the quadrature


21

q=0q

p=/2

q p=

qp=

q p= /2

q 0=

SPI

2PI

SPI

2PI

2PI stretching

squeezing

dN

Incre

ment

DD

xx

||/

co

h 3.0

2.0

1.0

0.6

2PI

SPI

3

2

1

0

||

Para

mete

rg

2

| | Parameterg2

2

1

0

4(

-)

DD

xx

||2

2 co

h

0

2

4

8

6

0 1 2 3 0 1 2q Parameter

a b

(i) (i)

(ii) (ii)

stretching

FIG. 6: Connection between physical quantities and the (γ, θ) parameters within the Schodinger cat state (86). a,Dependency of (i) δN and (ii) ∆x‖ on the γ parameter. The horizontal lines indicate the SPI (δN = 1) and 2PI (δN = 2) values.

The shaded area corresponds to quantum squeezing. b, Analysis of SPI (black lines) and 2PI (red lines) cat states: (i) |γ|2 and (ii) thequadrature fluctuations are plotted as function of θ. Results for both the stretching (γ = |γ|) and the squeezing (γ = i|γ|) cat are shown.All properties are obtained by solving Eqs. (92)–(93) and (96).

fluctuation along the laser’s β0 displacement is defined by ∆x‖ ≡ ∆x. In this situation, the γ = |γ| perturbationyields a stretching cat while γ = i|γ| defines a squeezing cat, compare Fig. 5.Figure 6a(ii) presents ∆x‖ for a stretching (dashed lines) and squeezing (solid lines) cat as function of |γ|2, using

θ = 0 (black curves), θ = π2 (red curves), and θ = π (blue curves). We see that the squeezing cat with θ = 0

(black line) displays the largest squeezing below the minimum uncertainly limit (shaded area). In this case, ∆x‖ dips

down to 0.67∆xcoh at |γ|2 = 0.64, which indicates substantial squeezing below the minimum uncertainty limit. Thissqueezing cat has ∆x‖ < ∆xcoh for all |γ| values while it asymptotically approaches the classical ∆x‖ → ∆xcoh limitfor elevated |γ| perturbations. For the same θ = 0, the stretching-cat perturbation produces a ∆x‖ (dashed lines)that always exceeds the classical limit, as it should. Due to these extremes, we have used the θ = 0 squeezing andstretching cat states in the Article to represent the overall capabilities of quantum-optical spectroscopy.The squeezing cat can also be realized for other θ parameters such as θ = π

2 (red line). However, θ = 0 produces thestrongest ∆x‖ squeezing features among all possible θ values. In fact, having θ above π

2 starts to produce quadraturefluctuations that exceed the classical limit for large enough γ. As the extreme case, θ = π (blue line) exceeds theclassical limit for all |γ|2. This follows because the γ → 0 limit produces a displaced Fock-state |1〉, as discussed inconnection with Eq. (91).

To demonstrate these features, we analyze in Fig. 6b(i) the |γ|2 parameter and (ii) the level of squeezing,

4(

∆x2‖ −∆x2coh

)

, as function of θ for the SPI (black lines) and 2PI (red lines) cat-state perturbations. We de-

termined |γ|2 numerically by evaluating the roots of Eq. (93) as function of θ. The results are completely independentof γ and the β direction. As a reminder, for the squeezing cat γ is perpendicular to β0 while the stretching cat isaligned along β0. Figure 6b(i) shows very clearly that θ = 0 produces a |γ|2 that approaches δN , especially for largeδN . Only the SPI sources show a significant θ dependence by dipping down to γ = 0 at θ = π. As shown by Eq. (91),this state corresponds to a displaced Fock state |1〉. The 2PI case only slight bows downward at θ = π.The corresponding quadrature fluctuations are shown in Fig.6b(ii) via 4(∆x2‖−∆x2coh). Only pure quantum sources

can make this quantity negative because this requires squeezing below the classically allowed limit ∆xcoh = 12 . This

classically forbidden regime is indicated by the shaded in the figure. We see that the squeezing-cat branch (γ = i|γ|)produces quantum squeezing for θ below roughly π

2 . The stretching-cat branch (γ = |γ|) is in the classically allowedregion for all θ values. We also see that SPI and 2PI produce qualitatively different squeezing patterns: the stretchingand squeezing-cat branch of 4(∆x2‖ − ∆x2coh) are connected at θ = π for SPI while these branches are completely

separated for the 2PI. This is a consequence of the lifting of the ∆x‖ degeneracy for the squeezing and streching catfor δN greater than one.For all cases, the maximum squeezing follows for θ = 0 for the squeezing-cat branch, as it should. For SPI, the


22

squeezing cat produces 4(∆x2‖ − ∆x2coh) = −0.40 while the stretching cat has 4(∆x2‖ − ∆x2coh) = 4.40. Therefore,

the stretching is roughly 11 times stronger in magnitude than the squeezing. This ratio of effects is clearly detectedin the analysis of Fig. 4 in the Article that directly monitors the squeezing-generated effects in the quantum-opticaldifferential spectrum.

E. General projection of the system response to Schrodinger cat states

To project the experimentally measured classical response R(|β|2) into the cat-state response, we insert P |β, γ, θ〉(β)of Eq. (89) into the CET formula (69). A straightforward integration eventually produces the quantum response

R|β,γ,θ〉QM =

NR

4∆x2R

C2

∑

J=0

(−1)JaR(J)N(

WJ

(

|Γ+|24∆x2

R

)

+WJ

(

|Γ−|24∆x2

R

)

+ 2e−|γ|2Re[

eiθWJ

(

Γ⋆−Γ+

4∆x2R

)])

. (97)

By comparing this with the classical CET response (68), we realize that the response to the cat-state is directlyconnected with the classically measured one via the transformation

RQM = N(

R(|Γ+|2) +R(|Γ−|2) + 2e−2|γ|2 Re[

eiθR(Γ⋆−Γ+)

]

)

. (98)

This contains the two classical responses R(|Γ±|2) evaluated at the ”ears” of the cat state. Since they are simplydisplaced classical responses, they can be understood semiclassically. The last contribution R(Γ⋆

−Γ+) clearly stemsfrom the ”whiskers” of the cat. This contribution implements an analytic continuation of the amplitude argumentinto the complex plane because Γ⋆

−Γ+ is no longer a real-valued quantity, compare Eq. (92). Since the analyticcontinuation of the response function cannot be understood classically, the cat state indeed produces true quantumcontributions to the system response, making them ideal for studying quantum-optical spectroscopy.

The analytic continuation of R(|β|2) into the complex plane requires that we know the exact mathematical formof this function. Obviously, experiments cannot directly provide such information without further considerationsbecause measurements access R(|β|2) only with finite accuracy and at a finite number of phase-space points. Themeasurements also include some noise that limits our capability to reconstruct the full mathematical form of R(|β|2).The presented CET result (97) avoids all these complications and allows us to project a set of classical measurementsinto true quantum sources. To phrase it differently, the CET projection yields a robust analytic continuation of theclassically measured data needed to access the quantum-optical response.

V. QUANTUM-OPTICAL SPECTROSCOPY WITH SEMICONDUCTOR QUANTUM WELLS

In our Article, we present measured properties of semiconductor quantum wells (QW) and used the CET methodto project the differential QW absorption where the laser perturbations are performed in the form of Schrodinger catstates. In this Section, we elaborate the experimental details further to present calibration and background informationto the work presented in the Article. We also deepen the analysis by studying general Schrodinger cat-state excitationsas well as excitations with strong-intensity quantum sources.

A. Experimental setup

We perform spectrally resolved transient absorption measurements on GaAs quantum wells (QWs). While manytransient absorption measurements have been performed in the past, we make quantitative measurements of the probeabsorption spectra in carefully calibrated absolute units. These measurements, combined with a characterization ofthe sample structure, allow quantitative comparison with a microscopic theory that can explore the effects of quantumstatistics on the quantum kinetics of many-body correlations, using the CET method.

More specifically, we study a sample consisting of 10 periods of 10-nm thick GaAs QWs separated by 10-nmAl0.4Ga0.6As barriers. To characterize the structure, we carefully measure the linear transmission and reflectionspectra of the unexcited sample using white light. The linear results are analyzed with a transfer-matrix computationthat includes all layers in the sample. We fine tune the refractive indices and thicknesses of the dielectric layers inthe model to reach a full agreement between linear theory and measurements for the energy range 1.4− 1.7 eV. Thisprocedure allows us to calibrate the nonlinear absorption experiments and determines all needed sample parameters,


23

which are fixed in all our subsequent quantitative theory-experiment comparisons, compare also Ref. [32]. We chooseas a central feature of study the 1s heavy-hole (HH) resonance of the QW at energy E1s =1.547 eV since it is wellseparated from other absorption resonances in the sample and has a strong oscillator strength, see Fig. 2a in theArticle. With the sample structure determined, we proceed to make quantitative measurements of the nonlinearabsorption.

Ti:Sapphiremode-locked laser

Spectralfilter

Adjustable ND filter

POL

POL

QWP

QWP

t delay

Rprobe

Tprobe

Spectrometers

QW sampleat 4K

Cryostat

IN

R

T

probe

pump

BS

BS

FIG. 7: Experimental setup for realization of quantum-optical spectroscopy using a GaAs quantum-well (QW) sample.Light from the mode-locked laser is split into pump and probe pathways. The pump passes through a spectral filter, and the power isadjusted via an adjustable neutral-density (ND) filter. The polarization of each path is controlled by a polarizer (POL) and quarter-waveplate (QWP). The incident, transmitted, and reflected (IN, T, R) pump powers from the QW sample are simultaneously measured usingphotodetectors. The calibrated transmitted and reflected probe spectra (Tprobe and Rprobe) are recorded with spectrometers.

The nonlinear optical probe response is recorded in a pump-probe configuration, as shown in Fig. 7. We usea mode-locked Ti:sapphire laser to produce the pump and the probe; such laser pulses are indeed nearly ideallyclassical, cf. Refs. [33, 34], which allows us to measure a complete set of CET input data in this setup. Our pumplaser spectrum is centered at E1s and spectrally narrowed to a half width half maximum (HWHM) bandwidth of 2.9meV to avoid exciting the electron-hole continuum or the light-hole resonance; pump spectrum is shown in Fig. 3ain the Article. We also have found the pump pulse duration to be 274 fs FWHM from an intensity autocorrelationwidth of 388 fs FWHM by assuming a Gaussian line shape. With a 4 nm bandwidth, the time-bandwidth product iscB = ∆τ∆ω = 0.51, which is close to the transform limit cB = 0.441 for a Gaussian pulse, which corroborates withnearly ideal laser characteristics.

This pump pulse is focused onto a 100µm excitation spot on QWs. After a delay τ , a low-intensity probe pulsearrives from a direction different from that of the pump in order to avoid direct pump-probe transfer. The weakprobe is focused to a 20 µm spot in the center of the pumped area to monitor only the spatially homogeneous partof the QW excitation. The pump and probe have opposite circular polarizations so that they are interacting withseparate exciton transitions according to the spin selection rules. We record the probe transmission T (ω) and reflectionR(ω) probabilities as functions of probe-photon energy ~ω for a matrix of time delays and pump powers. Throughsimultaneously calibrated measurements of the incident, transmitted, and reflected probe powers, we determine thetrue absorption α(ω) = 1 − T (ω) − R(ω). For growth purposes, the sample includes bulk GaAs that produces aspectrally flat background absorption of 28%. Thus, the QW absorption is αQW(ω) = α(ω)− 0.28 for this sample.The photon number of the pump pulse, N , is calibrated in a similar fashion to the probe via sampling pick-offs

as shown in Fig. 7. Through careful measurements of the incident, transmitted, and reflected pump powers, we caneven determine the number of photons absorbed by the QW from each pulse by considering reflection losses fromvarious surfaces, subtracting the bulk absorption background. This number was found to increase monotonicallywith N , as expected. For the strongest pump intensity, we complete saturate the excitonic 1s absorption. Inthe quantitative theory-experiment comparison in Ref. [32], we used resonant excitation resulting in an electron-hole density neh = 4 × 1010 cm−2 close to saturation. This means that the highest excitation intensities produceπ(50µm)2 × 4 × 1010 cm−2 ≈ 3 × 106 electron-hole pairs within the 100µm pump spot used. Therefore, the probeindeed monitors nonlinearities of a highly nontrivial many-body system, as stated in the Article.

Since the pump and probe have opposite polarization directions, pump coherences do not interfere with the probe.As a result, the probe absorption does not depend on the phase of the pump such that we can perform the CET scansin terms of the pump intensity N = |β|2, as discussed in Sec. III B. Therefore, the N of the pump pulse is variedto generate the CET scan that contains the relevant range of excitation conditions. By defining N of each pumppulse, we assign |β|2 = N such that the measured probe QW absorption can be stored as a two-dimensional matrixαQW(ω, β). We use this complete set of as input to our CET computations as outlined in Sec. III B.


24

B. Squeezing- and stretching-cat perturbations

Once we have recorded αQW(E, |β〉), shown in Fig. 2a in the Article, we can apply the CET steps (63)–(67)to construct the correlations aR(J) from the measured data. As we include aR(J) up to the C-particle level, wecan express the QW absorption analytically via Eq. (68). This step produces the CET form αCET(E, |β〉) of themeasured QW absorption. Figure 2b in our Article shows that the CET describes the nonlinear QW absorption withvery high accuracy. Using these results, we can project the quantum-optical QW absorption αQM(E) to any source.In particular, we can construct the quantum-optical differential absorption spectrum

∆α|β,γ,θ〉;|β0〉(E) ≡ α|β,γ,θ〉(E)− αCET(E, |β0〉) . (99)

between the classical laser |β0〉 and its perturbation with a Schrodinger cat state |β, γ, θ〉. The cat-state responseα|β,γ,θ〉(E) is constructed by inserting the CET coefficients aR(J) into Eq. (97). Technically, we determine thecoefficients aR(J) for each E separately. In other words, each fixed E produces its own CET projection into α|β,γ,θ〉(E)and αCET(E, |β0〉). Therefore, the probe-photon energy E simply parametrizes the analyzed data sets and the CETrelated to them.

The CET method is completely nonperturbative, i.e., we do not rely on the expansion (58) to obtain the differentialresponses. Therefore, we can equally well deduce the differential response to quantum-optical perturbations of anystrength. However, since Eq. (58) suggests that already weak quantum-optical perturbations can produce intriguingnew aspects to the spectroscopy, we mainly analyze single- and zero-photon increments. Here, Eq. (58) serves as anexcellent interpretation tool for the more general, nonperturbative computations.

To implement quantum SPI, we project the perturbation from the classical laser |β0〉 into the cat state (86),i.e. |β, γ θ〉, whose parameters are defined by Eqs. (92)–(93). Technically, we find the |γ| root for the δN = 1increment in Eq. (93), compare Fig. 6. The corresponding classical SPI perturbs the laser’s amplitude from β0 to β′

with the constrains |β′|2 = |β0|2 + 1, which implies only an infinitesimal increase β′ − β → 12|β0| in the displacement

for |β0| ≫ 1. Therefore, we can directly summarize the quantum-statistical changes in classical vs. quantum SPI:

δ〈B〉coh ≡ β′ − β0 → 12|β0| vs. δ〈B〉cat ≡ 〈B〉|β,γ,θ〉 − β0 = 0 ,

δ∆〈B†B〉coh = 0 vs. δ∆〈B†B〉cat ≡ ∆〈B†B〉|β, γ, θ〉 −∆〈B†B〉|β0〉 = δN = 1 ,

δ∆〈BB〉coh = 0 vs. δ∆〈BB〉cat ≡ ∆〈BB〉|β, γ, θ〉 −∆〈BB〉|β0〉 = γ2(1 + δf) . (100)

Since coherent states have no correlations, the classical SPI changes only the displacement. For the used SPI cat, thelaser’s displacement is not changed whereas the photon number as well as the two-photon absorption correlations aremodified according to Eq. (95).

Since SPI perturbations are weak for semiconductors, the differential QW response follows accurately from theexpansion (58). As we insert Eqs. (58) and (100) into Eq. (99), we find

∆αcoh(E) =∂αQW

∂N0vs. ∆αcat(E) =

∂αQW

∂N0+∂2αQW

∂2N0N0 Re

[

1 + γ2(1 + δf2)]

(101)

for the classical vs. cat-state SPI, respectively. To simplify the notation, we have identified the initial photon numberN0 = |β0|2 and used the property (96). We see that the cat state yields a new quantum contribution that is formallyas large as the classical term. The SPI differential spectra, therefore, contain a mixture of classical and quantumcontributions.

The quantum-statistical sensitivity can be accessed more directly by keeping the photon number constant while thelaser’s quantum-optical extension is perturbed, yielding the concept of zero-photon increment (ZPI). According toEq. (93), the displacement center β′

0 of the cat state then must satisfy |β′0|2 + δN = |β0|2. For the ZPI, the classical

changes vanish such that we find

δ〈B〉coh = 0 vs. δ〈B〉cat = β′0 − β0 → − δN

2|β0| for |β0| ≫ 1

δ∆〈B†B〉coh = 0 vs. δ∆〈B†B〉cat = δN

δ∆〈BB〉coh = 0 vs. δ∆〈BB〉cat = γ2(1 + δf2) (102)

for the classical vs. quantum ZPI, respectively. This follows directly from Eq. (101) because we only have to changethe cat-state’s displacement in order to produce ZPI perturbations to the laser. In all our ZPI computations, we set


25

Displacement Number correlation 2-photon correlation

classical

stretching cat

squeezing cat

SPI

ZPI

No

rma

lize

d c

ha

ng

e 0 0 0 0

000

FIG. 8: Schematic representation of single-photon increment (SPI) and zero-photon increment (ZPI). The scaled changesto the coherent displacement, the photon-number correlations, and the two-photon correlations are defined by Eqs. (101) and (102).

δN = 1 to get results comparable with SPI. Repeating an analogous computation as for the SPI, the correspondingdifferential ZPI absorption becomes

∆αcoh(E) = 0 vs. ∆αcat(E) =∂2αQW

∂2N0N0 Re

[

12N0

+ 1 + γ2(1 + δf2)]

(103)

for classical vs. quantum ZPI, respectively. Classically, the response is not changed while quantum response containsonly the quantum contributions. We also see that 1

2N0≪ 1 can be neglected for strong excitations with N0 ≫ 1.

Figure 8 schematically compares the quantum statistical perturbations needed to realize classical (while bars),stretching cat (yellow bar), and squeezing cat (blue bar) perturbations in SPI (upper part) and ZPI (lower part). It isnotable that changes in the coherent amplitude and number correlations add up to create classical contributions to thedifferential response whereas the number and two-photon correlations add up to produce the quantum contributionsto the differential response, compare Eqs. (100) and (102) with Eqs. (101) and (103).

C. Convergence of the CET projection

In this Section, we analyze how fast the CET approach converges as function of the cluster number used. Asdiscussed in Sec. VB, the first step of the CET projection produces the CET version αCET(E, |β〉) of the measuredQW absorption spectrum αQW(E, |β〉). Figure 9a shows the measured (shaded area) QW absorption together withits CET reconstruction using C = 5 (dashed line), C = 30 (blue line), C = 75 (red line), and C = 140 (gray line). Wenotice the clearly converging trend as C is increased. Furthermore, we see that the CET distributions are smoother,thus removing some of the experimental noise present in the measurement.

C=5

C=30 C=75

C=1

40

5

3075

140

CE

TD

evia

tion

10%

0%0 40 80 120Cluster number

BiX

BiX

-4 -2 0 2 4

Energy - [meV]E1s

QW

absorp

tion

0.2

0.1

0.0

-4 -2 0 2 4

Energy - [meV]E1s

Diffe

rential

absorp

tion [10

]-7 0.4

0.0

-0.3

a b

FIG. 9: Convergence of CET as function of cluster number. a, The measured QW absorption (shaded area) is compared withthe CET-constructed absorption using three different numbers of clusters. Here, we use the example where the measurement is donewith a pump laser having 19M photons on average. The inset presents the relative deviation of the CET construction from the measuredabsorption. b, The corresponding quantum-optical differential ZPI absorptions to a stretching cat perturbation. The vertical line indicatesthe spectral position of the biexciton resonance.


26

To quantify the convergence in more detail, we determine the normalized deviation

ǫCET(C) ≡∫

dE |αCET(E, C)− αQW(E)|∫

dE |αQW(E)| (104)

between the CET constructed and the measured QW absorption. The inset to Fig. 9a presents this deviation asfunction of the used number of clusters. We clearly observe good convergence and notice that cluster numbers aboveC = 75 yield virtually unchanged results. We also see that ǫCET(C) slowly approaches zero beyond C = 75. As shownin Ref. [24], the high-order clusters mainly describe rapidly changing contributions such as noise. In the present case,the truncation to C = 75 cluster produces an accurate description of the relevant physical effects while it additionallyfilters out the experimental noise to a large degree.

To see how fast the projected quantum-optical responses converges, we CET project the differential ZPI absorptionfor a stretching cat using the same CET data as in the αCET projection. Figure 9b presents a sequence of projected∆αZPI for the same numbers of clusters as in Fig. 9b. Only the lowest case (C = 5 dashed line) deviates from the restwhile a clear converging trend is observed between C = 30 (blue line) and C = 140 (gray line). We see that increasingC above 75 enhances the noise in the spectra, as argued above. Consequently, we find that C = 75 (red line) is anoptimal choice for the CET projection. This is therefore the value used in the analysis presented in our Article.

Here, we picked a strong-excitation value of the pump intensity where the signatures of the electron-hole complexstart to emerge in the spectrua. For a stretching cat, the signature is an additional dip below the biexciton resonance(vertical line in Fig. 9a). This feature is clearly visible and stable once the CET projection is accurate enough, i.e.,the deviation in CET of the initial spectrum is nearly saturated, see the inset to Fig. 9a.

D. Noise sensitivity in CET projection

Since any experiment is bound to have some noise, we obviously must quantify how well the CET projection filtersout this noise when projecting onto the quantum-optical responses. To test this thoroughly, we intentionally addrandom noise to the measured αQW(E, |β〉) for each (E, β) point recorded. For this purpose, we use a random-numbergenerator that produces uncorrelated noise in the interval [−∆ǫ, ∆ǫ]. Figure 10a presents the measured αQW(E, |β〉)(shaded area) together with the noise-added input data where we have used the noise amplitudes ∆ǫ = 0.01 (blackline), ∆ǫ = 0.03 (red line), and ∆ǫ = 0.06 (dashed line). The shown measurement corresponds to the pump excitationwith N = 19M photons. The other intensities are not shown but contain an equivalent amount of noise before theCET projection is applied.

Measured+0.01+0.03+0.06

+0.00+0.01+0.03+0.06

+0.00+0.01+0.03+0.06

BiX

BiX

BiX

-4 -2

-2

0

0

2

2

4

4

Energy - [meV]E1s

Energy - [meV]E1s

Diffe

rential

absorp

tion [10

]-7

0.4

0.6

0.4

0.2

0.2

0.10.0

0.0

0.0

-0.3

QW

absorp

tion 0.2

0.1

0.0

Differ

entia

l

CET-IN

CET-CET0

CE

Tdevia

tion

Input deviation

+0.01

+0.03 +0.0

6

a b

c

d

FIG. 10: Robustness of the CET projection against noise. a, The measured QW absorption (shaded area) is compared to threespectra with added random noise with ±0.01 (black line), ±0.03, and ±0.06 as maximum peak-to-peak amplitude. The corresponding CETprojected, b, QW absorption and, c, quantum-optical differential ZPI spectrum are presented for a stretching-cat perturbation. d, Therelative deviation between the CET constructed absorption and the measurement (shaded area) is shown as function of the relative noisein the original input data. The relative deviation in the CET-constructed QW (squares) and differential (circles) absorption is plottedusing the noisy vs. original input data.

The corresponding CET constructed QW absorption is shown in Fig. 10b. We use C = 75 in all cases to filter outthe noise and to reconstruct the physical QW absorption accurately. This choice suppresses the noise significantly


27

such that the CET-constructed distributions deviate much less from each other than the noisy input data. To quantifythe differences, we identify the normalized deviations

ǫin ≡∫

dE |αQW(E, ∆ǫ)− αQW(E)|∫

dE |αQW(E)| , ǫ(1)CET ≡

∫

dE |αCET(E, ∆ǫ)− αQW(E)|∫

dE |αQW(E)|

ǫ(2)CET ≡

∫

dE |αCET(E, ∆ǫ)− αCET(E, ∆ǫ = 0)|∫

dE |αCET(E, ∆ǫ = 0)| (105)

between the noisy and the measured input data. Here, ǫ(1)CET defines the corresponding deviations in the CET projection

and ǫ(2)CET it the deviation between the CET projection with and without noise. Figure 10d shows ǫ

(1)CET (shaded area)

and ǫ(2)CET (squares) as function of the input deviation ǫin. We see that both of them are roughly three times smaller

than the input deviation ǫin, which explicitly demonstrates how the CET suppresses the noise while accessing thephysically relevant part of the spectrum.

To truly test our approach, we CET project the differential ZPI absorption for the stretching cat using the noisydata as the input. The constructed ∆αZPI is shown in Fig. 10c for the corresponding inputs of Fig. 10a. We see thatthe projected ∆αcat nicely reproduces the noiseless reconstruction (shaded area) roughly up to a noise level ∆ǫ = 0.03.Remarkably, even the projection from the strongly noisy ∆ǫ = 0.06 input produces fairly good results where the mainqualitative features of the differential response are still clearly visible. In particular, the dip due to the electron-holecomplex is present regardless of the added noise level.

To quantify the accuracy of the CET projection further, we compute the deviation

ǫcat ≡∫

dE |∆αcat(E, ∆ǫ)−∆αcat(E, ∆ǫ = 0)|∫

dE |αcat(E, ∆ǫ = 0))| (106)

between the noisy and the noiseless projection of the differential spectra. The red circles in Fig. 10d show ǫcat asfunction of the input deviation ǫin. We see that ǫcat is only three times larger than ǫin nicely demonstrating that evenrather high levels of noise do not corrupt the result. All these tests convincingly verify that the CET indeed providesa robust method to access the quantum-optical responses for a wide range of systems.

Diffe

ren

tia

la

bso

rptio

n [

10

]-7 0.4

0.4

0.0 0.0

-0.2 -0.2

0 0-2 -20 0

2 2

p p

2p 2p

Cat-state

q

Energy -[meV]

E 1s

Energy -[meV]

E 1s

squeezing catsqueezing

cat

stretchingcat

stretchingcat

a b

FIG. 11: Differential QW absorption resulting from, a, SPI and, b, ZPI perturbations as function of θ. The CET analysisprojects ∆α(E) for 19M pump photons as function of the θ parameter in the cat state. The transparent gray surface corresponds to asqueezing cat and the colorful surface to the stretching cat. The red (blue) lines indicate the squeezing (stretching) cat cross-sections atθ = 0 and at the 1s resonance.

E. Squeezing dependence of the differential signal

Our Article presents quantum-optical differential spectroscopy results for the most extreme cases with θ = 0 in theSchrodinger cat state. To determine differential response for a more general cat state, we CET project ∆αcat usingSPI and ZPI perturbations around a laser with N = 19M photons. Figure 11 presents ∆αcat for a SPI and b ZPI


28

spectroscopy. The red-yellow surface corresponds to the stretching cat (γ = |γ|) and the black-shaded surface showsthe squeezing-cat (γ = i|γ|) results. The blue (red) curves indicate slices for the stretching (squeezing) cat taken atθ = 0 or at the 1s-exciton resonance. The computations have been performed as described in Sec. VB.

The ZPI differential spectrum reproduces very accurately the same X-shaped curves as function of θ as the quadra-ture fluctuations in Fig. 6b(ii). This shape is particularly clear for the cross sections with fixed E (blue and redline). Therefore, the θ dependence of the ZPI is determined by the θ-dependent modifications to the cat state’squadrature fluctuations in the laser’s displacement direction, as predicted by the perturbative Eq. (103). In otherwords, the spectral shape is independent of θ and its magnitude and sign follow entirely from the θ-dependent changein the quadrature fluctuations. Even though the CET computation includes effects to all orders, we see that the usedZPI introduces a weak perturbation for the studied semiconductor system such that the lowest-order perturbationexpansion (103) describes the quantum-optical changes accurately.

Equation (101) predicts that SPI mixes the classical and quantum response components. We see that especiallythe squeezing cat induces a ∆αcat whose spectral shape changes with θ. Therefore, one can apply θ to introducenontrivial changes to the SPI differential absorption. As a typical trend for both SPI and ZPI, θ = 0 produces thelargest quantum-optical modifications to the differential spectrum which is the reason why we exclusively use theθ = 0 cat state in the Article.

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