Relaxation of Radiation-Driven Two-Level Systems Interacting with a Bose-Einstein Condensate Bath

2 MB·

by  Vadim M. Kovalev

Relaxation of Radiation-Driven Two-Level Systems Interacting
with a Bose-Einstein Condensate Bath
Vadim M. Kovalev1,2∗ and Wang-Kong Tse3,4†
1Institute of Semiconductor Physics, Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russia
2Department of Applied and Theoretical Physics,
Novosibirsk State Technical University, Novosibirsk 630073, Russia
3Department of Physics and Astronomy, The University of Alabama, Alabama 35487, USA
4Center for Materials for Information Technology,
The University of Alabama, Alabama 35401, USA
(Dated: January 10, 2017)
7
We develop a microscopic theory for the relaxation dynamics of an optically pumped two-level
1
system (TLS) coupled to a bath of weakly interacting Bose gas. Using Keldysh formalism and
0
diagrammaticperturbationtheory,expressionsfortherelaxationtimesoftheTLSRabioscillations
2
are derived when the boson bath is in the normal state and the Bose-Einstein condensate (BEC)

n state. Weapplyourgeneraltheorytoconsideranirradiatedquantumdotcoupledwithabosonbath
a consistingofatwo-dimensionaldipolarexcitongas. WhenthebathisintheBECregime,relaxation
J of the Rabi oscillations is due to both condensate and non-condensate fractions of the bath bosons

7 forweakTLS-lightcouplinganddominantlyduetothenon-condensatefractionforstrongTLS-light
coupling. Our theory also shows that a phase transition of the bath from the normal to the BEC

] state strongly influences the relaxation rate of the TLS Rabi oscillations. The TLS relaxation rate
l
l isapproximatelyindependentofthepumpfieldfrequencyandmonotonicallydependentonthefield
a
strength when the bath is in the low-temperature regime of the normal phase. Phase transition of
h
the dipolar exciton gas leads to a non-monotonic dependence of the TLS relaxation rate on both
-
s the pump field frequency and field strength, providing a characteristic signature for the detection
e of BEC phase transition of the coupled dipolar exciton gas.
m
.
at I. INTRODUCTION strongly modifies the properties of the electronic subsys-
m tem, resulting in polariton-mediated superconductivity
- The dynamics of a quantum two-level system (TLS) is and supersolidity [13–17]. The topic of phase transition
d a topic of fundamental importance. Its sustained influ- of the exciton or exciton-polariton Bose system into the
n ence is evident in the continual interest in the dynamics BEC state is itself an intriguing topic that has garnered
o
ofspinorpseudospinsystemsrangingfromquantumop- much attention [18–22]. For dipolar exciton systems
c
[ tics [1, 2] to quantum information [3–5]. The spin-boson realized in GaAs double quantum well (DQW) struc-
model [6]captures theinteractionbetweenthe TLSand tures, the critical temperature to reach the condensate
1 its environment by a spin 1/2 degree of freedom coupled phaseisabout3−5K. Recentworkshavedemonstrated
v
linearly to an oscillator bath [7]. Despite the simplicity that double-layer structures based on transition metal
7
4 ofsuchamodel,itexhibitsarichvarietyofbehaviorand dichalcogenides (TMD) monolayers [23, 24] can further
8 describes a diverse array of physical systems and phe- push the transition temperature to ∼10−30K [25–27].
1 nomena [6, 8]. One of the quantum systems that is well Motivated by recent interest in hybrid fermion-boson
0 described by a TLS is the quantum dot (QD). Fueled systems and exciton-polariton physics mentioned above,
1. by interests in quantum information processing, coher- inthispaperweconsideraradiation-drivenquantumdot
0 ent optical control of quantum dots has seen substantial coupled to a dipolar exciton gas and study the influence
7 development in the past decade [9]. New functionalities ofthelatter’sBECphasetransitiononthedynamicsand
1 or tuning capabilities can be achieved with hybrid sys- relaxation of the QD states. The problem of TLS relax-
v: temsbyfurthercouplingQDstoothermaterials,suchas ation coupled to a fermionic bath transitioning to a su-
i nano-sizedcavity[10],graphene[11],andsuperconductor perconducting state was studied in the context of metal-
X
[12]. lic glasses [28]; however, the question of TLS relaxation
r Hybrid quantum systems comprising a fermion gas coupled to a bosonic bath transitioning to a BEC state
a
coupled to a boson gas constitute the condensed matter has not, up to the authors’ knowledge, been considered
analogueof3He-4Hemixtures. Insystemswhereanelec- before. Itisnoteworthytomentionthatourcurrentwork
trongasiscoupledwithexcitonsorexciton-polaritons,it is closely connected to the problem of a mobile impurity
wasrecentlypredictedthatthetransitionoftheexcitonic movinginaBEC[29],sincetherenormalizationofphysi-
subsystemtotheBoseEinsteincondensate(BEC)phase calpropertiesofamovingelectronthatstronglyinteracts
with the surrounding medium (polaron problem) can be
describedbyaquantumparticlecoupledwithabath [6].
OurtheoryconsistsofaTLSmodelingthegroundand
∗ [email protected] lowest excited states of the QD, which is coupled [30–
† [email protected] 32]toabathofweakly-interactingBosegasmodelingthe
2
dipolar exciton system. In contrast to the simple spin- the unitary transformation
boson model, the interaction between the QD and the
2D dipolar exciton gas in our system is described by a (cid:18)eiωt/2 0 (cid:19) (cid:18)e−iωt/2 0 (cid:19)
Sˆ= , Sˆ−1 = .
nonlinear couplingHamiltonian. WetaketheBosegasto 0 e−iωt/2 0 eiωt/2
be weakly interacting, exhibiting a normal phase as well (3)
as a BEC phase described by the Bogoliubov model [33–
35]. Our results demonstrate that the damping of the As a result, the Green’s function in the rotating frame,
Rabi oscillations of the TLS is highly sensitive to the Gˆ (t,t(cid:48))=Sˆ(t)Gˆ˜ (t,t(cid:48))Sˆ−1(t(cid:48)), is described by the equa-
0 0
phase transition of the bosonic bath.
tion
The rest of our paper is organized as follows. The
(cid:18) (cid:19)
second section is devoted to the development of general i∂ −ε −λ
t 0 Gˆ (t,t(cid:48))=δ(t−t(cid:48)), (4)
theoryfortherelaxationrateofanilluminatedTLScou- −λ∗ i∂ +ε 0
t 0
pledtoabosonicbath. Wethenapplyourgeneralresults
to the situation of a QD coupled with a dipolar exciton where ε = ∆−ω/2. To find the self-energies and life-
0
gas in the third section. Finally, in the fourth section times, we need the retarded and the lesser components
we present numerical results of the relaxation rates and of the non-equilbrium Green’s function. The retarded
discuss their behavior as a function of the optical pump Green’s function is derived as (see Appendix A),
field’sparameters. IntheAppendixwepresentdetailsof
our calculations. Aˆ Bˆ
GˆR(ε)= + , (5)
0 ε−Ω+iδ ε+Ω+iδ
where Ω=(cid:112)ε2+|λ|2 is the Rabi frequency, and Aˆ and
II. GENERAL THEORY 0
Bˆ are matrices defined as
A. Driven TLS and Rabi oscillations (cid:18) u2 uv (cid:19) (cid:18) v2 −uv (cid:19)
Aˆ= , Bˆ = , (6)
u∗v∗ v2 −u∗v∗ u2
First, we consider dynamics of isolated TLS system
under strong external electromagnetic field and describe with u2 = (1+ε /Ω)/2, v2 = (1−ε /Ω)/2 and uv =
0 0
thesystem’sresponseusingthenon-equilibriumKeldysh λ/2Ω. Eqs.(5)-(6)implythatnewquasiparticlesemerge
Greenfunctiontechnique. TheTLSHamiltonianisgiven fromthelight-mattercouplingthatrenormalizestheorig-
by inal TLS states into dressed states with energies ±Ω.
Aˆ and Bˆ are the projection operators to these dressed
Hˆ˜ (t)=(cid:18) ∆ λe−iωt (cid:19), (1) states.
0 λ∗eiωt −∆ It is instructive to recover the result for Rabi oscilla-
tions using the above retarded Green’s function Eq. (5).
The TLS wave function at a latter time t is obtained by
where ±∆ are the energies of the upper and lower
propagating the initial time (t=0) wave function,
states of the TLS, and quantities with an overhead caret
(ˆ) symbol denotes a matrix quantity. The interaction
ψ (t)=[Sˆ−1(t)GˆR(t)Sˆ(0)] ψ (0)
Hamiltonian with the electromagnetic field is written i ij j
here in the Rotating Wave Approximation (RWA). λ is =(cid:20)Sˆ−1(t)(cid:90) dεGˆR(ε)e−iεt(cid:21) ψ (0), (7)
the interaction matrix element and ω the frequency of 2π j
ij
the electromagnetic field. It is also assumed in Eq. (1)
that the wavelength of the electromagnetic field is much whereψ arethewavefunctionsoftheTLSupper(i=1)
i
larger than the geometrical size of the TLS so that the and lower (i = 2) levels. Here Sˆ−1(t)GˆR(t)Sˆ(0) ≡
field is uniform on our scale of interest. In this work, Gˆ˜R(t,0) is the retarded Green’s function in the labora-
we denote quantities in the laboratory frame and the ro-
tory frame. When only the lower level is initially occu-
tatingframe,respectively,withandwithoutanoverhead
pied, ψ (0) = 1 and ψ (0) = 0. Using Eqs.(3)-(6), the
tilde. 2 1
transition probability to the upper level is then given by
The dynamics of the TLS is described by the time-
oofrdmeroetdioGnreen’sfunctionGˆ˜0(t,t(cid:48))satisfyingtheequation |(cid:104)ψ2+(0)ψ1(t)(cid:105)|2 =(cid:12)(cid:12)(cid:12)(cid:12)Ωλ sin(Ωt)e−iωt/2(cid:12)(cid:12)(cid:12)(cid:12)2
(cid:18) i∂t−∆ −λe−iωt (cid:19)Gˆ˜ (t,t(cid:48))=δ(t−t(cid:48)). (2) = 2|λΩ|22 (1−cosΩt), (8)
−λ∗eiωt i∂ +∆ 0
t
which is the Rabi oscillations [36].
To remove the explicit time dependence, it is convenient The lesser Green’s function can be expressed in terms
to transform this equation to the rotating frame using of the distribution functions n of the upper and lower
±Ω
3
dressed states as On the other hand, Eq. (9) gives the density matrix in
(cid:104) (cid:105) the basis of the dressed quasiparticles as follows
Gˆ<(ε)=−n GˆR(ε)−GˆA(ε)
ε (cid:90) dε
(cid:104) (cid:105) fˆ(t)=−iGˆ<(t,t)=−i Gˆ<(ε),
=2πin Aˆδ(ε−Ω)+Bˆδ(ε+Ω) 2π
ε
(cid:104) (cid:105) =Aˆn +Bˆn . (17)
=2πi Aˆn δ(ε−Ω)+Bˆn δ(ε+Ω) . (9) Ω −Ω
Ω −Ω
We can determine n by comparing the expressions of
Ω
Note that nΩ + n−Ω = 1. Assuming the radiation is the density matrix in Eq. (17) and Eq. (15). The 11
turned on adiabatically, we can obtain n±Ω in the fol- element, for instance, gives f11 = u2nΩ+v2(1−nΩ) or
lowing. The density matrix fˆ in the original basis of n =(f −v2)/(u2−v2), from which we determine
Ω 11
TLS upper and lower levels satisfies the kinetic equation
(see Appendix A): 1
n = [1∓sgn(ε )]. (18)
±Ω 2 0
∂fˆ
+i[Hˆ ,fˆ]=0, (10) It follows that n takes on the values 0 or 1 depending
∂t 0 Ω
(cid:18) (cid:19) on whether the light frequency ω is smaller or larger,
f f
fˆ= 11 12 , (11) respectively, than the energy difference 2∆ of the TLS.
f f
21 22
where the subscripts 1,2 respectively denotes the origi-
B. Coupling to bosonic bath and TLS self-energies
nal(i.e.,unrenormalizedbylight)upperandlowerlevels
of the TLS, and Hˆ = Sˆ(t)Hˆ˜ (t,0)Sˆ−1(0) is the Hamil-
0 0 In the absence of bath coupling, the TLS is described
tonian in the rotating frame. Writing the Hamiltonian
as Hˆ = σˆ ·B /2, we can define an effective magnetic by bare Green’s functions with a vanishing level broad-
0 0 ening. To focus on the effects of bosonic bath coupling,
field B = 2λ e −2λ e +(2∆−ω)e that drives the
0 R x I y z weignoretheeffectsofspontaneousandstimulatedemis-
TLS pseudospin degrees of freedom, where λ are the
R,I sion due to electrons’ coupling to light. The only damp-
real and imaginary parts of λ and e the unit vectors
x,y,z ing effects on the TLS dynamics, once coupled to the
along the x,y,z directions. Then, decomposing the den-
bosonic bath, will be due to interlevel transitions caused
sity matrix fˆas fˆ= C +S ·σˆ/2, the kinetic equation
by absorption or emission of the bosons. To analyze the
can be written as a Bloch equation:
TLS-bathcoupling,weaddtothebareTLSHamiltonian
∂S Eq. (1) the TLS interaction term with the bath and the
+S×B =0. (12)
∂t 0 bath Hamiltonian
(cid:18) (cid:19)
From the definition of fˆ in Eq. (11), we can relate W11[ϕ] 0 +Hˆ [ϕ]. (19)
0 W [ϕ] bath
the distribution functions in the two representations as 22
S = f − f and S(+) ≡ S + iS = 2f∗ = 2f .
z 11 22 x y 12 21 Here the first term is TLS-bath coupling Hamiltonian
With the laser field switched on adiabatically, the opti-
wherethematrixelementsdescribetheinteractionofthe
cal response follows adiabatically the driving field and is
upperandlowerlevelswiththebathbosons. Bothterms
thereforestationaryintherotatingframe,i.e.,∂/∂t=0.
in Eq. (19) are functionals of the quantum field ϕ, which
Before laser is turned on, the TLS initial state is in the
describes the dynamics of bath degrees of freedom. The
lower level so that S(t = 0) = −e . Since |S| is a con-
z structure of the bath Hamiltonian depends on whether
stant of motion, this implies that |S(t)|=1 for all times
it is in the normal or Bose-condensed phase and will be
t. Here we focus on the regime without population in-
specified later on. We assume short-range interaction
version, so that S = f −f is always less than zero.
z 11 22 between the TLS and the bosonic bath
We obtain S as
(cid:90)
sgn(2∆−ω)2λ∗ W [ϕ]=g dr|ψ (r)|2|ϕ(r,t)|2, (20)
S(+) =− , (13) ii i i
(cid:112)
(2∆−ω)2+4λ2
|2∆−ω| with the coupling constant gi. The form of this interac-
Sz =−(cid:112) . (14) tion contains ϕ2 and is markedly different from the con-
(2∆−ω)2+4λ2
ventional coupling to a phonon-like bath, which is linear
Using f +f =1, we also find the density matrix fˆin in ϕ [6]. Anticipating further application of our general
11 22
theory to consider a QD interacting with a 2D exciton
the original basis of the TLS upper and lower levels:
gas, we note that W in Eq. (20) takes into account the
(cid:34) (cid:35) ii
1 |2∆−ω| most important direct contribution to the Coulomb in-
f = 1− , (15)
11 2 (cid:112)(2∆−ω)2+4λ2 teractionenergybetweenelectronsintheQDandthe2D
exciton gas.
sgn(2∆−ω)2λ To elucidate the influence of the bosonic bath on the
f =− . (16)
12 (cid:112)
(2∆−ω)2+4λ2 TLS dynamics, we use the diagrammatic perturbation
4
theory. Within this approach the retarded Green func-
tion can be found from the Dyson equation,
GˆR =GˆR+GˆRΣˆRGˆR, (21)
0 0
FIG.1. FeynmandiagramforTLSself-energywhenthebath
is in the normal phase. Double blue line represents the TLS
where the bare Green’s function GˆR0 is given by Eq. (5). Green function, dashed lines the TLS electron-bath interac-
The solution of Eq. (21) gives the interacting Green’s tion,andsolidblacklinestheGreenfunctionsofthenormal-
function as follows state bath particles.
1 (cid:104) (cid:105)
GˆR = Λ GˆR0 −det(GˆR0)σy(ΣˆR)Tσy , (22) 1 =−ImTr(cid:104)AˆΣˆR(Ω)(cid:105), (24)
2τ
u
1 (cid:104) (cid:105)
=−ImTr BˆΣˆR(−Ω) , (25)
2τ
l
where Λ = 1−Tr(GˆRΣˆR)+det(GˆRΣˆR), (ΣˆR)T is the and neglected the shift of the levels due to the real part
0 0
matrix transpose of ΣˆR, and σ is the Pauli matrix. We of self-energy. To obtain expressions of the relaxation
y
arrive at the following form of the Green’s function (see times,weapplythediagrammaticperturbationtheoryin
Appendix B) the leading order of the TLS-bath interaction potential
andaccountforthelowest-ordernon-vanishingdiagrams
for the self-energy. The explicit form of such diagrams
depends on whether the bath is in the normal or the
condensate state. First we consider the normal state.
Aˆ Bˆ
GˆR((cid:15))≈ + , (23)
(cid:15)−Ω+i/2τu (cid:15)+Ω+i/2τl 1. Bath in the normal state
In the normal state of the bath, we assume that the
bosons are non-interacting with a kinetic energy E =
p
|p|2/2m ≡ p2/2m and chemical potential µ. To lowest
where we have introduced the relaxation times τ of order in the TLS-bath interaction, the self-energy dia-
u,l
the upper (subscript u) and lower (subscript l) dressed gram is shown in Fig. 1. Detailed calculation of the self-
quasiparticles energy is presented in the Appendix C. The result reads
(cid:20)
ΣˆR(ω)=(cid:88) Mˆ AˆMˆ (1−nΩ)[nB(ξp)−nB(ξp+k)]+nB(ξp+k)[1+nB(ξp)]
k −k ω−Ω+E −E +iδ
p p+k
k,p
(cid:21)
(1−n )[n (ξ )−n (ξ )]+n (ξ )[1+n (ξ )]
+Mˆ BˆMˆ −Ω B p B p+k B p+k B p , (26)
k −k ω+Ω+E −E +iδ
p p+k
where ξ =E −µ is the energy of the bath bosons ren- ing expression into Eqs. (24)-(25) we find the relaxation
p p
dered from the chemical potential, n (ξ)=[exp(ξ/T)−
B
1]−1 is the Bose distribution function, and
(cid:18)g (cid:82) dreikr|ψ (r)|2 0 (cid:19)
Mˆk = 1 0 1 g (cid:82) dreikr|ψ (r)|2 , (27)
2 2
Taking the imaginary part of Eq. (26), integrating over
the angle between k and p, and substituting the result-
5
times interaction. In the low-energy, long-wavelength limit
1 m (cid:90) ∞ (cid:34)(cid:90) ∞ α fu(p)pdp Ep (cid:28) 2g0nc the elementary excitations compr(cid:112)ise sound
2τu = (2π)2 0 dk k/2 (cid:112)pk2Ω−(k/2)2 (28) qisutahnetas,ouwnitdhvaelodcisiptye.rsIinonth(cid:15)epB≈ossep-cwonhdeernessed=statge0,nmc/omst
 of the particles are in the condensate but there are also
+(cid:90) ∞ (cid:113) βkf−uΩ(p)pdp , noncondensate particles, due to both interaction and fi-
|k/2−2mΩ/k| p2−(k/2−2mΩ/k)2 nite temperature effects (thermal-excited particles). All
threefractionsofparticlescontributetorelaxationtimes
(cid:34)
1 m (cid:90) ∞ (cid:90) ∞ γ fl (p)pdp of TLS. We consider the quantum limit T (cid:28) sp when
2τl = (2π)2 0 dk k/2 (cid:112)kp2−−Ω(k/2)2 (29) tbheedrmevaelloexpceidtaftoironTsa=re0n.oItnimthpeoprtraensetnatnddiltuhteetbhoesoornycgaans

(cid:90) ∞ β fl(p)pdp limit,thedensityofthenoncondensateparticlesissmall,
+ (cid:113) k Ω . and one can neglect the interaction term due to fluctu-
|k/2+2mΩ/k| p2−(k/2+2mΩ/k)2 ations of the condensate density and the noncondensate
density. Thus, the contribution to relaxation times due
Here we have introduced the coefficients to the condensate and noncondensate particles can be
calculated independently.
αk =Tr(cid:16)AˆMˆkAˆMˆk∗(cid:17)=(cid:12)(cid:12)g1(Mk)11u2+g2(Mk)22v2(cid:12)(cid:12)2,
(cid:16) (cid:17)
β =Tr BˆMˆ AˆMˆ∗ =|uv|2|g (M ) −g (M ) |2,
k k k 1 k 11 2 k 22
γk =Tr(cid:16)BˆMˆkBˆMˆk∗(cid:17)=(cid:12)(cid:12)g1(Mk)11v2+g2(Mk)22u2(cid:12)(cid:12)2, (a)
(30)
and
(b)
fu(p)=fl (p)=n (ξ )[1+n (ξ )],
Ω −Ω B p B p
fu (p)=n [n (ξ )−n (ξ +2Ω)]
−Ω Ω B p B p FIG.2. FeynmandiagramsforTLSself-energywhenthebath
+nB(ξp+2Ω)[1+nB(ξp)], is in the BEC phase. Type (a) describes the condensate par-
fl(p)=θ(E −2Ω){n [n (ξ )−n (ξ −2Ω)] ticles contribution, Σ¯c, and (b) is due to the non-condensate
Ω p −Ω B p B p particles, Σ¯n. Double blue line represents the TLS Green
+nB(ξp−2Ω)[1+nB(ξp)]}, (31) function and dashed lines the TLS electron-bath interaction.
RedlinesdenotethenormaloranomalousGreenfunctionsof
the non-condensate particles, and the zigzag lines stand for
√
2. Bath in the condensed state n corresponding to the condensate particles.
c
WenowconsiderthebathtobeintheBosecondensed The self-energy diagrams in the lowest order with re-
phase and obtain the TLS self-energy. The elementary spect to the TLS-bath coupling are depicted in Fig. 2.
excitationsoftheBose-condensedsystemareBogoliubov Fig. 2(a) corresponds to the contribution to the self-
quasi-particles. An explicit form of the dispersion law energy from the condensate particles and describes vir-
of Bogoliubov excitations depends on the model used tual processes in which a condensate particle is scat-
to describe the system of interacting bosons. In the tered by a TLS electron through an intermediate non-
case of small particle density an appropriate theoreti- condensate state. Fig. 2(b) corresponds to the contri-
cal model is the Bogoluibov model of weakly-interacting bution from the non-condensate particles and describes
Bose gas. In the framework of this model, the dis- polarization of the non-condensate particles induced by
persion law of elementary excitations has the form of theTLSelectrons. Theseself-energycontributionsdueto
(cid:15) = (cid:112)E (E +2g n ), where n is particle density condensateparticlesΣˆcR andnon-condensateΣˆnR parti-
p p p 0 c c
in the condensate and g the strength of inter-particle cles have the following form (see Appendix C)
0
(cid:20) (cid:18) (cid:19)
ΣˆcR(ω)= nc (cid:88)k Mˆ AˆMˆ 1−nΩ + nΩ
2ms k −k ω−Ω−(cid:15) +iδ ω−Ω+(cid:15) +iδ
k k
k
(cid:18) (cid:19)(cid:21)
1−n n
+Mˆ BˆMˆ −Ω + −Ω , (32)
k −k ω+Ω−(cid:15) +iδ ω+Ω+(cid:15) +iδ
k k
6
ΣˆnR(ω)= (ms2)2 (cid:88)(cid:34)MˆkAˆMˆ−k (cid:18) 1−nΩ + nΩ (cid:19)
2 (cid:15) (cid:15) ω−Ω−(cid:15) −(cid:15) +iδ ω−Ω+(cid:15) +(cid:15) +iδ
p p+k p p+k p p+k
k,p
Mˆ BˆMˆ (cid:18) 1−n n (cid:19)(cid:35)
+ k −k −Ω + −Ω . (33)
(cid:15) (cid:15) ω+Ω−(cid:15) −(cid:15) +iδ ω+Ω+(cid:15) +(cid:15) +iδ
p p+k p p+k p p+k
Similar steps leading to Eqs. (28)-(29) yields
1 = 1 =n πnc (cid:88)kβ δ(−2Ω+(cid:15) ), (34)
2τc 2τc Ω2ms k k quantum
u l k dot
1 1
= (35)
2τn 2τn spacer
u l
=n π(ms2)2 (cid:88) βk δ(2Ω−(cid:15) −(cid:15) ).
Ω 2 (cid:15) (cid:15) p p+k
p p+k
k,p double
quantum
Explicit expressions for relaxation times depend on the
well
shape of the wave functions of the TLS upper and lower
statesthroughthematrixelementsEq.(27). Inthenext
FIG.3. SchematicofthecoupledQD-dipolarexcitonssystem.
sectionweproposeandanalyzeanexperimentalsetupin
Thequantumdotispositionedthroughadielectricspaceron
which explicit expressions of the relaxation time can be
top of a double quantum well that hosts the dipolar exciton
obtained. gas.
III. APPLICATION TO COUPLED dius. For simplicity, we assume that there is no particle
QD-DIPOLAR EXCITON BATH tunneling between the QD and the DQW so that the ex-
change contribution to the interaction potential is neg-
Applying our theory developed in the previous sec- ligible. This can be guaranteed using a large-band gap
tions, we consider the nanostructure depicted in Fig. 3. dielectric spacer as a substrate between the QD and the
A double quantum well (DQW) with closely separated DQW. The direct contribution to the interaction poten-
electron-doped and hole doped wells realizes a 2D gas of tial between the electron-hole pair in the QD and the
indirect excitons (also called dipolar excitons). A self- excitons in the DQW is given by
assembled QD is positioned above the DQW and is ir-
(cid:90) (cid:90)
radiated with a frequency close to the lowest exciton en- U = dr dr(cid:48)(cid:0)|χ (r)|2−|χ (r)|2(cid:1)|ϕ(r(cid:48))|2u(r−r(cid:48)),
h e
ergyoftheQD.Intheelectron-holerepresentationofthe
QD,thelowerquantumstatedescribestheunexcitedQD (36)
state (i.e. vaccum) |vac(cid:105), while the lowest excited QD e2 e2
state describes the state |eh(cid:105) of an excited electron-hole u(r−r(cid:48))= (cid:112) − (cid:112) ,
ε (r−r(cid:48))2+a2 ε (r−r(cid:48))2+(a+d)2
pairuponirradiation[37–39]. Withashiftinenergythat
(37)
does not affect the system’s dynamics, we can assume
thatthesetwoQDstatescorrespondexactlytothelower
whereχ (r)aretheelectron(e)andhole(h)wavefunc-
and upper TLS levels in Eq. (1) with ∆=(E +E )/2, e,h
e h tions in the QD, ϕ(r) is the QW exciton center-of-mass
where E are the lowest energies of electrons and holes
e,h wave function, ε is an effective dielectric constant tak-
in the QD (corresponding to the energies at the conduc-
ing account of the dielectric environment of the spacer
tionandvalencebandedges),andλrepresentsthedipole
and the DQW, a and d are the distance of the QD to
matrix element weighted by the electron-hole pair enve-
the DQW surface and the separation between the posi-
lope wave function integrated over all space.
tive charges and negative charges of the dipole layer (see
Letusdiscusstheexcitongasmodelweuseforourcal-
Fig. 3). The Fourier transform of u(r) yields
culations. The indirect excitons have a dipole moment p
ianrethmeooduetl-eodf-paslanriegiddirdeciptioolne omfotlheecuDleQsWth.aTt haereexfrceiteontos u(k)= 2πe2 (cid:0)1−e−kd(cid:1)e−ka. (38)
εk
move on the DQW plane as described by the center-of-
mass motion of the dipoles, a valid assumption as long Typical values of the wave vector k here are determined
as the dipole’s internal degrees of freedom are not ex- by the bath excitons of DQW. At temperatures above
cited. The exciton density n is assumed to be small the condensation temperature T > T , the bath in the
ex c
so that n a2 (cid:28) 1, where a is an exciton Bohr ra- normal state, the DQW exciton energy is of the order of
ex B B
7
√
T andthusthewavevectork ∼p = 2mT,wheremis to(cid:126)2/(2md2)],wecankeepuptothezerothorderinΩ/T
T
theexcitonmass. AtT =0,thebathisinthecondensed withF (p,p )≈6πp4 andfu (p)≈fl(p)≈n (ξ )[1+
± 0 −Ω Ω B p
phase,thetypicalvalueofthebathexcitationmomentum n (ξ )]inEqs.(41)-(42). TherelaxationratesforΩ(cid:28)T
B p
is of the order of p ≤ ms. Hereafter we assume the then read
distance of the QD from the DQW as well as the inter-
1 1 (cid:16) ε (cid:17) (cid:16) (cid:17)
welldistanceintheDQWtobesufficientlysmallsothat = 1± 0 Li eµ/T , (44)
τ τ Ω 2
p a,p d (cid:28) 1 and (ms)a,(ms)d (cid:28) 1. This allows us to u,l T
T T
simplify the expression Eq. (38) assuming that kd,ka(cid:28)
where the +,− signs apply for τ respectively, and
u,l
1. Thus we have
1 3U2m4T3(a2 −a2)2
2πe2d = 0 h e , (45)
u(k)≡U0 = ε , u(r)=U0δ(r). (39) τT 4π(cid:126)9
and Li (x) is the polylogarithm function of order 2. In
Under this approximation the interaction Eq. (36) be- 2
the opposite limit of low temperatures T (cid:28) Ω, we keep
comes contact-like,
up to the zeroth order in T/Ω. Then the respective first
(cid:90) terms in Eqs. (41)-(42) drop out, and we have fu (p)≈
U =U dr(cid:0)|χ (r)|2−|χ (r)|2(cid:1)|ϕ(r)|2. (40) −Ω
0 h e n n (ξ ) and fl(p)≈n θ(E −2Ω)n (ξ −2Ω) in the
Ω B p Ω Ω p B p
second terms. The result is
In our coupled QD-exciton gas system, since the lower
level is the vacuum, we have in Eqs.(19) and (20) that 1 = 1 = 1 n π2 (cid:18)λ(cid:19)2 (cid:126)2nex, (46)
the coupling constants g = U , g = 0 and their re- τ τ τ Ω 3 T mT
1 0 2 u l T
spective matrix elements W = U, W = 0. The
11 22
where n is the exciton density in the bath. We have
QD is described by a two-dimensional system of elec- ex
restored(cid:126)intheexpressionsfortherelaxationrateshere
trons and holes confined by a parabolic potential. For
and in the following section.
strongly confined QDs where the QD size L is small
compared to the Bohr radius of the electron-hole pair,
the lowest-energy electrons and holes are characterized
√ B. Relaxation times for Bose-condensed bath
bythewavefunctions[38]χ (r)=exp(−ρ2/2a2)/(a π),
i i i
where a is an electron i=e or a hole i=h characteris-
i
tic length determined by the QD confinement potential. Straightforward but cumbersome integration in
Using these wavefunctions, we find the matrix element Eqs. (34)-(35) results in the following expressions for re-
(Mˆ ) =exp[−(ka /2)2]−exp[−(ka /2)2]. laxationtimesduetothecondensateandnon-condensate
k 11 h e
particles in the bath
1 1 |λ|2n U2 (cid:104) (cid:105)2
A. Relaxation times for normal-state bath 2τc = 2τc =nΩ 4(cid:126)3mcs40 e−(Ωah/s)2 −e−(Ωae/s)2 ,
u l
(47)
With ka (cid:28) 1, we perform the integration over k in
i and
Eqs. (28)-(29) and obtain the following expressions for
the relaxation times 1 1 |λ|2m2sU2
= =n 0 (48)
21τu =+(m2(πmα2)π0β2)0(cid:90)20(cid:90)∞∞F+F(+p(,p0,)pn0B)f(ξ−upΩ)([1p)+pdnpB,(ξp)]pdp(41) ×2τ(cid:34)unF(√a22τhΩl√na2h/s)Ω+16Fπ2(√(cid:126)a42eΩΩ√2a2e/s) −2F((cid:112)(cid:112)aa2h2h++aa2eΩ2e /s)(cid:35),
0
1 mγ (cid:90) ∞ where F(x) is the Dawson integral
= 0 F (p,0)n (ξ )[1+n (ξ )]pdp
2τ (2π)2 − B p B p (cid:90) x
l mβ 0(cid:90) ∞ F(x)=e−x2 et2dt.
+ 0 F (p,p )fl(p)pdp, (42) 0
(2π)2 − 0 Ω
0
where F (p,p ) = 6π(p4 ±p2p2 +p4/6)θ(p2 ±p2) with IV. DISCUSSION
± 0 0 0 0
p2 =4mΩ, and
0
Simple analysis of Rabi oscillations given above ac-
α =U2(a2 −a2)2u4/16,
0 0 h e counting for the finite values of relaxation times yields
β =U2(a2 −a2)2|uv|2/16,
0 0 h e
γ0 =U02(a2h−a2e)2v4/16. (43) |(cid:104)ψ2+(0)ψ1(t)(cid:105)|2 = 2|λΩ|22 (1−cosΩt)e−t(1/2τu+1/2τl)
TosimplifyEqs.(41)-(42)weconsidertwolimitingcases. |λ|2 (cid:16) (cid:17)2
AtlargetemperaturesT (cid:29)Ω[whilestillsmallcompared + e−t/2τu −e−t/2τl . (49)
4Ω2
8
Wefocusourdiscussiononthelowtemperatureregime
T (cid:28) Ω. First we note that the relaxation rates for the
upperandlowerlevelscoincidebothinthenormalphase 3
[Eq. (46)] and in the Bose-condensed phase [Eqs. (47)-
(48)]. With relaxation times for the upper and lower
levels being equal, Eq. (49) is simplified to 2
|λ|2
|(cid:104)ψ+(0)ψ (t)(cid:105)|2 = (1−cosΩt)e−t/τ,
2 1 2Ω2 1
where 1/τ = 1/τ = 1/τ , with 1/τ given by Eq. (46)
u l
in the normal phase and by 1/τc+1/τn from Eqs. (47)-
(48) in the BEC phase. Secondly, the relaxation rates 0.5 1 1.5 2 2.5 3
are all proportional to the distribution function n of
Ω
the dressed quasiparticle states given in Eq. (18). Since
nΩ =1onlyiftheupperdressedquasiparticlestate+Ωis FIG.5. Relaxationrate(cid:126)/τn duetonon-condensateparticles
occupied and vanishes otherwise, finite relaxation of the (normalized by ∆) versus frequency detuning y. Red (dot-
TLS Rabi oscillations occurs only when the pump field dashed), blue (dashed) and black (solid) lines correspond to
frequencyexceedstheTLSenergyleveldifferenceω ≥2∆ λ=0.1meV, 0.2meV and 0.3meV respectively.
[40]. In our following discussion, therefore, we focus on
the regime ω ≥ 2∆. In the normal phase, we note that
the low-temperature relaxation rates Eq. (46) are inde-
6
pendentofthedrivingfrequencyandincreasesmonoton-
ically with the TLS-light coupling as λ2. In the BEC 5
phase, we find that the relaxation rates Eqs. (47)-(48)
exhibitstrongnon-monotonicdependenceonthedriving 4
frequencythroughtheRabifrequencyandtheTLS-light
3
coupling.
2
1
5
4 2 4 6 8
3
FIG. 6. Relaxation rates due to condensate and non-
2 condensate particles versus TLS-light coupling λ/∆ at fre-
quency detuning y = 0.01. Solid (black) line corresponds to
1 (cid:126)/τc and dashed (red) line to (cid:126)/τn, respectively.
0.5 1 1.5 2 2.5 3
constant takes the range of values λ ∼ 0.1 − 10meV.
For the dipolar exciton gas, we take d = 10nm, n =
c
FIG. 4. Relaxation rate (cid:126)/τc due to condensate particles 1010cm−2, ε = 12.5 and m = me+mh = 0.517m0 typ-
ical for GaAs DQW structures. To provide an estimate
(normalized by ∆) versus frequency detuning y. Red (dot-
dashed), blue (dashed) and black (solid) lines correspond to for the inter-particle interaction g0, we assume a sim-
λ=0.1meV, 0.2meV and 0.3meV respectively. ple point-charge treatment of the dipolar excitons, and
the exciton-exciton interaction potential takes the form
√
Below, we proceed to analyze the numerical depen- [44] g(r) = (2e2/ε)(1/r − 1/ r2+d2). Fourier trans-
dence of the relaxation rates on the driving frequency form of g(r) then gives g(k) = (4πe2/εk)[1−exp(−kd)]
and TLS-light coupling in the BEC phase. For the and the coupling constant g0 ≡ g(k = 0) = 4πe2d/ε.
TLS, we take the following parameters ∆ = 500meV, The Bogoliubov speed of sound s is thus also fixed from
(cid:112)
me = 0.067m0 and mh = 0.45m0 (m0 is the electron s= g0nc/m.
mass) typical for GaAs-based QDs [41–43]. The char- Forconvenience,wedisplaythefrequencyωintermsof
acteristic lengths of the hole and the electron wavefunc- thedimensionlessfrequencydetuningy =(ω−2∆)/(2∆).
tions are taken as a = 2nm and a ≈ a (cid:112)m /m = Figs. 4-5 show the relaxation rates (cid:126)/τc and (cid:126)/τn as
h e h h e
2.23a . With the dipole matrix element of the QD a function of y for relatively small values of λ ∼
h
∼10−100Debye(1Debye=3.3×10−30Cm)andoptical 0.1meV. Wefindthatbothrelaxationratesbehavenon-
field strength 0.1 − 10MVm−1, the TLS-light coupling monotonically as a function of detuning, reaching max-
9
face acoustic waves (SAW). Under phase transition, the
SAW attenuation effect and the SAW-exciton drag cur-
rent become strongly modified, allowing one to detect
the BEC phase transition using acoustic spectroscopy.
Ontopoftheforegoing, ourfindingsinprincipleprovide
anewstrategytodetecttheBECphasetransitionofthe
dipolar exciton gas. While the QD’s relaxation rate dis-
plays only a monotonic linear dependence on the light
intensity when the exciton gas is in the normal phase,
it becomes strongly non-monotonic as a function of both
the pump field’s frequency and intensity once the exci-
ton gas is in the BEC phase. Thus, by monitoring the
RabioscillationdynamicsoftheQD,thenormalandcon-
densed phases of the exciton gas can be distinguished by
the dependence of the relaxation rate on the frequency
and intensity of the driving field.
FIG. 7. Three-dimensional plot of relaxation rates (cid:126)/τc and
(cid:126)/τn as a function of TLS-light coupling λ/∆ and frequency
detuningy. Grey(red)surfacecorrespondsto(cid:126)/τc andblack
(blue) surface to (cid:126)/τn, respectively. V. CONCLUSION
Toconclude,wehavedevelopedatheoryfortherelax-
imum values at y ∼ 0.01 and then becoming exponen- ationofopticallypumpedtwo-levelsystemscoupledtoa
tially suppressed at larger values of y. (cid:126)/τc is more bosonicbathusingthenonequilibriumKeldyshtechnique
strongly suppressed than (cid:126)/τn. Secondly, we observe and the diagrammatic perturbation theory. To elucidate
that, for the present values of λ ∼ 0.1meV, the con- the effects of bath phase transition, we have considered
densate and non-condensate fractions contribute to the the cases when the bosonic bath is in the normal state
relaxation rate by the same order of magnitude, with andintheBose-condensedstate. Wethenapplyourthe-
(cid:126)/τc exceeding (cid:126)/τn. This trend is maintained until λ ory to study the scenario of an illuminated quantum dot
reaches∼1%of∆(correspondingto5meV), when(cid:126)/τc coupled to a dipolar exciton gas. The condensate and
startstodropsignifcantlyfasterthan(cid:126)/τn. Fig.6shows non-condensatefractionsofthebathparticlescontribute
bothquantitiesplottedversusλ/∆atafrequencydetun- to the relaxation rate by variable proportions depending
ing y =0.01, from which we observe that (cid:126)/τc decreases on the value of pump field amplitude. When the pump
muchmoreabruptlythan(cid:126)/τn. Whenλisincreasedbe- fieldisweak,bothfractionscontributebyaboutthesame
yond ∼ 0.02∆, it is seen that (cid:126)/τn now overtakes (cid:126)/τc. orderofmagnitude;whileforstrongpumpfield,thenon-
Forλvaluesbeyond0.03∆, (cid:126)/τc hasdroppedessentially condensate fraction becomes the dominant contribution.
to zero and the non-condensate fraction constitutes the Our findings also show that the phase transition of the
dominant contribution to relaxation. dipolarexcitongastotheBECregimeresultsinastrong
Although the relaxation rates vanish expectedly when dependence of the relaxation rate on the optical pump
λ = 0, they do not vanish at zero frequency detuning field. The relaxation rate then exhibits a strong non-
y =0, as one might conclude by inspecting Figs. 4-5. To monotonic behavior, reaching a maximum and then be-
examine more fully the behavior of (cid:126)/τc and (cid:126)/τn, we coming exponentially suppressed as a function of both
plottheminFig.7inthefullrangeofλandy. Aty =0, the pump field’s frequency and amplitude. Such a non-
both relaxation rates become small only when λ (cid:28) ∆; monotonicdependencecouldinprincipleserveasasmok-
in addition (cid:126)/τc also become small when λ (cid:38) 0.03∆. inggunfordetectingBECphasetransitionofthecoupled
Around λ ∼ 0.01∆, both relaxation rates as a function dipolarexcitongas. Finally,wepointoutthatdespiteour
of y reach maximum at y =0. focus on dipolar exciton gas in this work, the theory we
have developed is also applicable to other types of Bose
Conventionally, the BEC phase transition of a dipo-
gas,suchas2Dexciton-polaritons[47],magnons[48]and
lar exciton gas is detected using optical spectroscopy. In
cold atoms [49].
the BEC phase, the excitons or exciton-polaritons are
described by a single coherent wave function and emit
light coherently. The resulting luminescence peak be-
comesmuchnarrowerincomparisonwiththatinthenor-
VI. ACKNOWLEDGMENTS
mal phase, signaling formation of the condensate state.
Another way to detect the BEC phase transition has
beentheoreticallysuggestedrecently[45,46]. BECphase V.M.K. acknowledges the support from RFBR grant
transition strongly influences the non-equilibrium prop- #16−02−00565a. W.K. acknowledges the support by
erties of a dipolar exciton gas driven by an external sur- a startup fund from the University of Alabama.
10
VII. APPENDIX Solving this matrix equation yields the retarded Green’s
function in Eq. (5):
A. Non-Equilibrium Green’s Functions
(cid:18) (cid:19)
1 ε+ε λ
GˆR(ε,T)= 0
Because of the time-dependent perturbation from 2Ω λ∗ ε−ε0
(cid:20) (cid:21)
light, we employ the Keldysh formalism to calculate the 1 1
× −
Green’s function and distribution function of the sys- ε−Ω+iδ ε+Ω+iδ
tem. Following established routes in non-equilibrium 1 1
Green’sfunctionformalism,theleft-multipliedandright- =Aˆ +Bˆ , (56)
ε−Ω+iδ ε+Ω+iδ
mulitplied Dyson equations for the contour-ordered
Green’s function Gc are with Aˆ,Bˆ defined in Eq. (6).
Now from the contour-ordered Dyson’s equations
G−1Gc =1+ΣcGc, (50)
0 Eq. (51) and applying Langreth’s rule, then subtracting
GcG−01 =1+GcΣc. (51) the two equations, we get G−1G<−G<G−1 =0,
0 0
We are interested in the Green’s function of the TLS (cid:32) (cid:33) (cid:32) (cid:33)
∂Gˆ< ∂Gˆ<
under irradiation, therefore the self-energy Σ due to in- i −Hˆ Gˆ< − −i −Gˆ<Hˆ =0,(57)
teraction with the bath is set to zero. In the rotating ∂t 0 ∂t(cid:48) 0
frame, we already find TransformingintotheWignercoordinates,weobtainthe
kinetic equation
(cid:18)i∂ −(cid:0)∆− ω(cid:1) −λ (cid:19)
Gˆ−01(t,t(cid:48))= t −λ∗ 2 i∂t+(cid:0)∆− ω2(cid:1) . (52) i∂Gˆ< −(cid:104)Hˆ ,Gˆ<(cid:105)=0 (58)
∂t 0
First we derive the retarded Green’s function. Applying
Langreth’srules[50]tothetwoequationsinEq.(51)and The density matrix f(t) is given by the equal-
summing them together, we have Gˆ−01GˆR +GˆRGˆ−01 = time Keldysh Green’s function fˆ(t) = −iGˆ<(t,t) =
2δ(t−t(cid:48)), −iGˆ<(T =t,τ =0), which satisfies
(cid:32) (cid:33) (cid:32) (cid:33)
∂GˆR ∂GˆR
i −Hˆ GˆR + −i −GˆRHˆ =2δ(t−t(cid:48)). ∂fˆ
∂t 0 ∂t(cid:48) 0 ∂T +i[Hˆ0,fˆ]=0. (59)
(53)
We transform the time variables t,t(cid:48), into the Wigner
B. Quasiparticle Lifetimes
coordinates with the average time T = (t + t(cid:48))/2 and
relative time τ =t−t(cid:48). Eq. (53) becomes
We start from Eq. (22) in the main text
∂GˆR (cid:110) (cid:111)
i2 − Hˆ ,GˆR =2δ(τ). (54) 1 (cid:104) (cid:105)
∂τ 0 GˆR = Λ GˆR0 −det(GˆR0)σy(ΣˆR)Tσy .
Performing Fourier transformation with respect to τ
gives From Eqs. (5)-(6) it can be easily evaluated that
det(GˆR)=1/(ε2−Ω2+iδ). UponsubstitutionofEq.(5),
(cid:110) (cid:111) 0
2εGˆR− Hˆ ,GˆR =2. (55) the first term of Eq. (22) can be written as follows
0
GˆR Aˆ
0 =
Λ ε−Ω−Tr(AˆΣˆR)−Tr(BˆΣˆR)(ε−Ω)/(ε+Ω)+det(ΣˆR)/(ε+Ω)
Bˆ
+ . (60)
ε+Ω−Tr(AˆΣˆR)(ε+Ω)/(ε−Ω)−Tr(BˆΣˆR)+det(ΣˆR)/(ε−Ω)
In the vicinity of the poles we have Since ΣˆR ∼ 1/τ where τ is the quasiparticle lifetime,
1/τ (cid:28) Ω is satisfied for our perturbative calculations.
GˆR0 ≈ Aˆ (61) We see that the last terms in the denominators above
Λ ε−Ω−Tr(AˆΣˆR)|ε=Ω+det(ΣˆR)|ε=Ω/(2Ω) det(ΣˆR)|ε=±Ω/(2Ω) ∼ (1/τ) × 1/(Ωτ) are a factor of
Bˆ 1/(Ωτ)smallerthan1/τ andthuscanbeneglected. Next
+ .
ε+Ω−Tr(BˆΣˆR)| −det(ΣˆR)| /(2Ω)
ε=−Ω ε=−Ω