Lasing at arbitrary frequencies with atoms with broken inversion symmetry and an engineered electromagnetic environment

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by  Michael Marthaler

Lasing at arbitrary frequencies with atoms with broken inversion symmetry and an
engineered electromagnetic environment
Michael Marthaler,1 Martin Koppenh¨ofer,1 Karolina Sl owik,1,2 and Carsten Rockstuhl1,3
1Institut fu¨r Theoretische Festko¨rperphysik, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany
2Instytut Fizyki, Uniwersytet Mikol aja Kopernika, Torun´, Poland
3Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
(Dated: January 8, 2016)
With the purpose to devise a novel lasing scheme, we consider a two level system with both
a transversal and longitudinal coupling to the electromagnetic field. If the longitudinal coupling
is sufficiently strong, multi-photon transitions become possible. We assume furthermore that the
electromagneticenvironmenthasaspectrumwithasinglesharpresonance,whichservesasalasing
6
cavity. Additionally, the electromagnetic environment should have a very broad resonance around
1
a frequency which differs form thesharp resonance. Weuse thepolaron transformation and derive
0
a rate equation to describe the dynamics of such system. We find that lasing at the frequency of
2
thesharp modeis possible, if theenergy differenceof theatomic transition is similar tothesum of

n the frequencies of both peaks in the spectral function. This allows for the creation of lasing over a
a large frequencyrange and may in perspective enable THz lasing at room temperature.
J

7 PACSnumbers: 42.50.-p,81.07.Ta,42.60.-v,73.20.Mf

]
h
I. INTRODUCTION [23, 24]. In this work, we consider a situation where an
p
artificialtwo-levelatomis coupledto anelectromagnetic
-
nt The standard source of coherent light from infrared environmentwithanadjustableandwellcontrolledspec-
a to visible frequencies is the laser [1]. A laser requires, tral density.
u first, population inversion in an active medium to cre- A particular example would be to couple an artificial
q ate photons and, second, a cavity to allow for stimu- atom to a metallic nanostructure. These metallic nanos-
[
lated emission [2]. For an active medium consisting of tructuresrespondresonantlytolightatdifferentfrequen-

1 natural atoms, all non-diagonal matrix elements of the cies, where they may sustain localized surface plasmon
v dipole coupling between electromagnetic field and atom polaritons[25]. Plasmonicnanostructures,therefore,can
1 vanish because of the inversion symmetry of the atomic be used to tailor the local density of states of the elec-
1
Coulomb potential [3]. tromagnetic modes [26–28]. In principle, it is possible to
5
However, artificial atoms with broken inversion sym- createaplasmonicstructurewitharelativelysharpmode
1
0 metries, like superconducting qubits [4], quantum dots and an additional spectrally broad, dissipative contribu-
. [5,6],ormolecules[7]havebeenstudiedaswell. Forsuch tion. This requires the use of a plasmonic nanostruc-
1
artificial atoms the typical Jaynes-Cummings Hamilto- ture possessing, e.g. a trapped or a Fano type resonance
0
nian contains an additional σ -type coupling, which is [29, 30], and a broad continuum with an electric dipolar
6 z
1 oftencalledlongitudinalcouplingincontrasttothestan- response. Thisisthekindofstructurewehaveinthefol-
: dard transversal σ coupling. Lasing devices based on lowing in mind. We want the dissipative contribution in
v x
solidstate qubits havebeen studied [6, 8–10]anda large the spectral density to have a maximum at a frequency,
i
X body of work exists on lasing with self-assembled quan- which is the difference between the level splitting of the
r tum dots [11–14] and lasing with organic molecules [15]. artificial atom and the frequency of the sharp peak. We
a Forsuperconductingdevices,non-linearitiesbasedonthe willshowthatinthiscaseitispossibletogeneratelasing
Josephson effect have been proposed as a way to cre- atthe frequency ofthe sharpmode. By flexiblytailoring
ate a non-linear coupling to the electromagnetic field the resonance frequency of the electromagnetic environ-
[8, 16, 17]. In experimentally realized lasing devices, us- ment,thisallowsforlasingatalargerangeoffrequencies.
ing superconducting qubits [18–20] or gate defined dou- If the upper level of the two level system is excited
bledots[21],theσ couplingbetweenartificialatomand beyond the thermal occupation, lasing becomes possible
z
cavity field is present but has not yet been studied. throughthefollowingprocess: Thelongitudinalcoupling
Artificialatoms,whichhaveanadditionallongitudinal allowsamultiphotonprocess,aphotoniscreatedinboth,
coupling to the electromagnetic field, can show a partic- the sharp mode and in the broad part of the spectral
ularlyrichbehavioriftheycoupletoasharpcavitymode density. The energy for this process comes from the two
and additionally to a broad spectral density, which can level system, which relaxes into the ground state. The
be treated as a reservoir in equilibrium [22]. The longi- sum of the energies of the two photons has to be similar
tudinal coupling allows then for resonant multi-photon to the energy of the two level system. The excitation of
transitions. It has been shown that, depending on the the broad spectral density can be assumed to relax fast,
spectral function of the bath, effects exist, which can while the photon inthe sharpmode has a longlive time.
enhance stimulated emission of such an artificial atom Thisprocesscreatesaneffectivepopulationinversionbe-
2
tweenthestateofthesystemwithaphotoninthe broad II. THE SYSTEM
spectral density and the state without this photon. This
initiatesthelasingprocess. Thedescriptionofthisentire Wewillconsideratwolevelsystem,coupledtoasingle
process is at the heart of this publication. sharp mode and to an additional reservoir
To do so, we use in the following the polaron trans-
1
formation to bring the Hamiltonian that describes the H = ∆Eσ +(σ +σ cotθ)(x+x¯)+H +H . (4)
z x z s d
2
entire systeminto a form,whichallowsus to understand
multi-photon transitions directly by looking at possible Here, σ are the Pauli matrices acting on the artificial
i
transitionmatrixelements. The broadbackgroundspec- atom. Thesizeofthelongitudinaldipolematrixelement
trumcanbeusedtodressthecouplingbetweenthesharp is characterized by the angle θ. We can transform this
mode andthe twolevelsysteminawaythatallowsfora Hamiltonian using the polaron transformation
convenient calculation of transition rates. From this we
U =eiσz(p+p¯). (5)
can understand the properties of photon emission and
absorption into and from the sharp mode. After the unitary transformation we have
The energy distribution of the electromagnetic modes
H = U†HU
is described by the Hamiltonians H for the sharp mode P
s
and Hd for the broad mode spectrum, = H0+Hg+Hd,
1
H = ∆Eσ +Ωa†a, (6)
H = Ωa†a, (1) 0 2 z
s
H = ν b†b . Hg = σ+e−i2p¯e−ipxe−ip+h.c. , (7)
d Xk k k k Hd = (cid:0)(σ+e−i2pe−ip¯x¯e−ip¯+h.c.)(cid:1). (8)
Here, we have divided the Hamiltonian in a part which
Thebroadeningofthesharpmodewillbeincludedusing
containsthesystem(6),andtwopartswhichcontainthe
a standard master equation approach.
coupling between system and reservoir, (7) and (8) re-
Theannihilation(creation)operatorofthesharpmode
spectively. We will treat the coupling perturbatively in
is given by a (a†). The broad spectrum is described by
the lowest order. The coupling terms create transitions
the annihilation (creation) operators bk (b†k), which an- between the eigenstates of the system, which consists of
nihilate (create) an excitation in mode k. thetwolevelsystem,describedby∆Eσ /2andthesharp
z
The considered two level system is characterized by a mode described by Ωa†a. The eigenstates are given by
level splitting ∆E. It is coupled to the sharp mode with the product base of the two level system and the photon
thestrengthg,andadditionallytoeachofthemodeskin states of the sharp mode, o,n . The variable o can take
| i
the broad spectrum with the strength g . We define the the values + and with σ = . The photon
k z
angleθ suchthatthetransversalcouplingisproportional states are numbere−d by n, w|i±thia†a±n|±=i nn . There-
to sinθ and the longitudinal coupling is proportional to fore,thestateshavetheenergy(∆Eσ| /i2+Ωa|†ai) ,n =
z
|± i
cosθ. Towritethe totalHamiltonianinacompactform, ( ∆E/2+Ωn) ,n .
± |± i
we define a quasi-coordinate x and quasi-momentum p The transition rate from the state o,n to the state
for the sharp mode o′,m is given by | i
| i
x = gsinθ(a†+a), (2) Γo,n→o′,m =Γdo,enc→ayo′,m+Γpo,hn→o′,m, (9)
gcosθ where we divided the rate into two components. The
p = i Ω (a†−a). first rate Γdo,enc→ayo′,m corresponds to an expansion of the
term (8) and the second rate Γph corresponds to
o,n→o′,m
Similarly for the broad spectrum we define a quasi- an expansion of (7), respectively.
coordinate x¯ and quasi-momentum p¯ Thephysicalmeaningofthesetworatesiswelldefined
forθ =π/2,i.e. forapurelytransversalcoulping. Inthis
x¯ = sinθ gk(b†k+bk), (3) case we have Γdo,enc→ayo′,m ∝ δn,m. This means Γdo,enc→ayo′,m
Xk corresponds to a decay rate of the two level system, but
g leavesthe photonnumber inthe sharpmode unchanged.
p¯ = icosθXk νkk(b†k−bk). For θ =π/2 the rate Γpo,hn→o′,m obeys the condition
Γph δ , Γph δ . (10)
In the next section we will discuss the transformation +,n→−,m ∝ m,n+1 −,n→+,m ∝ m,n−1
of the basic Hamiltonian of our system. Then we will This means Γph contains the transition
o,n→o′,m
discuss how to derive transitionrates between the eigen- ,n ,n+1, which is the transfer of an excita-
↑ →↓
state of the system. In the last section we will discuss tion of the two level system into the photon field which
how to create lasing within our model. is crucial for lasing.
3
In the following we will discuss the relevant noise cor- Wewilldiscusslaterthevariousphysicaleffectsthatcan
relators, which are connected to the transition rates. be part of Γ.
To calculate Γpo,hn→o′,m we need an explicit form for the Using our definition for SP(ω), the spectral function
correlator e−2ip¯(t)e2ip¯(0) while for Γdecay we need becomes
h i o,n→o′,m
(e−ip¯x¯e−ip¯)(t)(eip¯x¯eip¯)(0) . With these results at hand S (ω) = exp( [η +η ]) (17)
hwe will calculate the transiitionrates and show that they P −ηm+ηn − 2Γ
correspond to lasing. + − .
× n!m! Γ2+(ω (n m)ω )2
Xn,m − − L
The spectralfunction has many peaks. Eachpeak corre-
A. Polaronic coupling to the reservoir
sponds to a process where the two level system changes
its state, emits or absorbs a photon to/from the sharp
If a system couples to a reservoir via an operator of
mode, and an additional m photons to/from the broad
the form e−2ip¯, we have to consider the noise correlator
spectraldensity. We willlaterdiscussparticularfeatures
CP(t)=he−2ip¯(t)e2ip¯(0)i, (11) which appear for a resonance condition, ∆E−Ω≈ωL.
wheretheaveragingisperformedbytracingoverthebath
degreesoffreedom. Thiscorrelatoriswellknown[31]and B. Broadened linear coupling to the reservoir
can be written in an explicit form as
Ifasystemcouplestoareservoirviaanoperatorofthe
CP(t)=eπ4R0∞dωJω(ω2)hcoth2kωBT(1−cosωt)−isinωti, (12) form e−ip¯x¯e−ip¯, we have to consider the noise correlator
where J(ω) is the spectral density of the reservoir. We C (t)= (e−ip¯x¯e−ip¯)(t)(eip¯x¯eip¯)(0) . (18)
D
h i
will later discuss in more detail how this correlator cor-
responds to transition rates in the system. The spec- To find an explicit form for this correlator, we use the
tralfunctioncanbeconnectedtotheeffectiveimpedance generating function
Z(ω) of the reservoir by
F = TeiRdt′(µ(t′)x¯(t′)−iλ(t′)p¯(t′)) , (19)
h i
J(ω)=2πωReZ(ω)/R , (13)
K
whereT isthetimesortingoperator. Usingthisfunction,
where RK =h/e2 is the resistance quantum. we can show that the relaxation correlator is given by
The impedance, which we want to consider, is the
impedance of an electromagnetic mode with frequency d2F
C (t)= .
ω . We want this mode to be very broad, such that we D − dµ(0)dµ(t)(cid:12) λ(t′) =(δ(t−t′+η)+δ(t−t′−η))
L (cid:12) −(δ(t′+η)+δ(t′+η))
canassumethatitalwaysstaysinequilibrium. However, (cid:12)µ(t′) =0
(cid:12)
we will first recall the result for a perfectly sharp mode (20)
and we will later introduce the broadening phenomeno-
logically. The impedance for a sharp mode is given by The explicit form of the generating function reads
2ReRZ(ω) =ǫC(δ(ω−ωL)+δ(ω+ωL)), (14) F = exp(cid:20)−Z dt1Z dt2F˜(t1,t2)Θ(t1−t2)(cid:21) , (21)
K
F˜ = (µ(t )µ(t ) x¯(t )x¯(t )
where ǫ is a coupling constant. In this very particular 1 2 1 2
C h i
case given by Eq. (14) this would correspond to ǫC = −iµ(t1)λ(t2)hx¯(t1)p¯(t2)i
gk2/νk,wherethewavevectorkisfixedbytherelationship iλ(t1)µ(t2) p¯(t1)x¯(t2)
to the frequency ω =ν . − h i
L k λ(t )λ(t ) p¯(t )p¯(t ) ) .
ForthisexampletheexplicitformofEq. (12)isknown − 1 2 h 1 2 i
and can be written as [31] We can apply Eq. (20) to this form of the generating
function and get
ηmηn
C (t) = e−(η++η−) + −e−i(n−m)ωLt (15)
P m!n! C (t) = x¯(t)x¯(0) +4 p¯(t)x¯(0) 2 (22)
Xm,n D h i h i
4ǫ cos2θ 1 (cid:2) e−2ip¯(t)e2ip¯(0) . (cid:3)
η = C ± . ×h i
± ωL e±ωL/kBT −1 We now wish to consider again a situation with an
To model the mode not as a sharp feature but to give impedance given by Eq. (14). As before, we do not
it a certain width Γ, we now introduce the broadened want a sharp mode, but it should have a certain width
spectral function Γ. Therefore, we define the spectral function
∞ ∞
S (ω)= dtC (t)eiωte−Γ|t|. (16) S (ω)= dtC (t)eiωte−Γ|t|. (23)
P P D D
Z Z
−∞ −∞
4
To write the resulting spectral function in a compact course suppress all other transitions. Additionally, the
form we first define matrixelements ne±ipxe±ip n+m and ne±2ip n+m
h | | i h | | i
scale like (gcosθ/Ω)m for gcosθ/Ω 1. Therefore, we
Γω2 tan2θ η ≪
S (ω) = L − (24) will consider the approximations
D,eff 2 (cid:18)Γ2+(ω ω )2
L
−
η η2 ne±2ip n 1 , ne±2ip n+m 0, (28)
+ + + + h | | i≈ h | | i≈
Γ2+(ω+ω )2 Γ2+(ω+2ω )2 n 1e±ipxe±ip n gsin2θ√n ,
L L
h − | | i≈
η2 2η η ne±ipxe±ip n 1 gsin2θ√n .
+ − − + . h | | − i≈
Γ2+(ω 2ω )2 − Γ2+ω2(cid:19)
L
− Using this approximation we can simplify the rates to
This spectral function together with Eq. (17), gives us
the relaxation spectral function Γph = Γ =g2(n+1)sin2θS (δω) ,(29)
+,n→−,n+1 +,n P
SD(ω) = exp(−[η++η−]) (25) Γp−h,n→+,n−1 = Γ−,n =g2nsin2θSP(−δω) ,
η+mη−nSD,eff(ω (n m)ωL). Γd±e,cna→y∓,n = SD(±∆E) ,
× n!m! − −
Xn,m
where we have introduced the detuning δω =∆E Ω.
−
Here we see that the total spectral density S consists
D
of multiple peaks, which repeat with distance ω . This
L
means relaxation is always strong for ∆E = m¯ω , with
L D. Self-consistent photon creation rates
m¯ = 1, 2,....
± ±
In the previous section we discussed the transition
C. The transition rates rates in the lowest order approximation. In this section
we want to go somewhat further. We will focus on the
photon creation rate. As can be seen by our approxima-
We can now write down the transition rates as calcu-
tion shown in Eq. (28), in general the transition matrix
lated by a second order expansion of H and H . The
g d
elementfortransitionsthatincreaseordecreasethenum-
photon creation rate is given by
ber of photons by one, grows with the total number of
Γph = ,mσ e±ipxe±ip ,n 2 (26) photons. This is the effect knownas stimulatedemission
±,n→∓,m |h∓ | ∓ |± i| (absorption), which is a key ingredient of the lasing pro-
∞
dtC (t)ei((n−m)Ω±∆E)te−Γ|t| cess. However, calculating a rate in the lowest order is
P
× Z−∞ onlypossibleaslongastherelevantnoisecorrelatorhasa
= ,mσ e±ipxe±ip ,n 2 decayratelargerthentheprefactoroftherate[32,33]. It
∓
|h∓ | |± i|
seems clearthat this conditionwill be violatedfor larger
S ((n m)Ω ∆E).
P
× − ± photon numbers. Therefore, we will use a self-consistent
The decay rate can be written as approachwhichwillguaranteeconvergenceforourrates.
We can make the rate defined in Eq. (26) a self-
Γdecay = ,nσ e±2ip ,m 2 (27) consistent equation, by including the rate itself as an
±,m→∓,n |h∓ | ∓ |± i|
∞ effective broadening of the energy,
dtC (t)ei((n−m)Ω±∆E)te−Γ|t|
D
×Z−∞ ∞
= ,mσ∓e±2ip ,n 2SD((n m)Ω ∆E). Γs+,n = g2(n+1)sin2θZ dtCP(t)eiδωte−ΓT|t|/2e−Γ|t|
|h∓ | |± i| − ± −∞
∞
Wewanttopointoutthatincontrasttostandardgolden Γs = g2nsin2θ dtC (t)e−iδωte−ΓT|t|/2e−Γ|t| (30)
rule rates, we have here the additional broadening Γ. −,n Z P
−∞
As discussed above, it allows us to model phenomeno-
logically the width of the dissipative contribution of the with ΓT =Γs+,n+Γs−,n and the detuning δω =∆E−Ω.
electromagnetic environment, while we can still use the Additionally, instead of solving the full-self-consistent
formalanalyticalresult,whichcanbederivedforasharp equation, we approximate the rates in the exponents by
mode. the lowest order results, ΓT = Γ+,n +Γ−,n. The self-
But our rates still have the standard form of a tran- consistent approach used here has previously been dis-
sition matrix element multiplied by the spectral func- cussed in relation to decoherence rates of superconduct-
tion evaluated at the relevant frequency. An interesting ing qubits [34, 35], transport across an Anderson quan-
factaboutthistransitionmatrixelementsisthattheyal- tum dot [36] and is often called self-consistent Born-
lowformulti-photontransitions. However,ifaresonance approximation[37].
conditionapplies, whichfavorsthe transitionwhich only We want to consider the spectral function of a single
changes the photon number by one photon, this will of mode(14). Ifwecalculatetherates(30)forthisspectral
5
function, we get an infinite sum of Lorentzian functions pump, which is used to excite the two level system. In
principle, it is possible to achieve Γ0 = Γ0. We would
Γs+,n = g2(n+1)sin2θe−(η++η−) (31) like to have a small decay rate Γdeca+y a−nd, therefore,
+,n→−,n
ηm′ηn′ 2Γ+Γ the energy splitting ∆E should not be resonant with a
+ − T ,
m′!n′! (Γ+Γ /2)2+(δω (n′ m′)ω )2 multiple of ωL.
mX′,n′ T − − L Dissipation in the oscillator is described by
Γs = g2nsin2θe−(η++η−) (32)
−,n κ
η+m′η−n′ 2Γ+ΓT . Ldissρ = 2 (cid:0)2aρa†−a†aρ−ρa†a(cid:1). (36)
m′!n′! (Γ+Γ /2)2+(δω+(n′ m′)ω )2
mX′,n′ T − L Here, the Hamiltonian Hsys and the Lindblad operator
do not mix the off-diagonal and diagonal components of
We will consider a case where we have δω ω . In this
case, the photon creation rate Γs is rel≈ativLely large, the reduced density matrix. Therefore,the systemis de-
+,n scribedbythediagonalcomponentsandevolvesstochas-
independent of temperature. We see that the photon
tically. We discuss the distribution probability of pho-
absorptionrateΓs issuppressedaslongasω k T.
−,n L ≫ B tons,
We want to emphasize that we only need ω k T to
L B
≫
create lasing, but Ω, the frequency of the lasing cavity
ρ = σ,nρσ,n , (37)
can be much smaller than temperature. n h | | i
σX=↑,↓
In the limit of large photon number n we find
Γnph,+ ≈ Γnph,− ≈ S ( δω)4+S (δω) . (33) anWd aevweraangtetnoudmebrievreoafnpheffoteocntisvehneiq=uaPtionnnfρonr.the prob-
P P
− ability distribution of the number of photons in the os-
Therefore,aswewouldexpectforlargephotonnumbers, cillator ρ = ρ . We do this by tracing out the
n σ σ,n
absorption and emission are equal. degrees of freePdom of the two-level system in the equa-
tion of motion given by Eq. (34). Using the relation
ρ =ρ +ρ andthe assumptionthatthe time scales
n ↑,n ↓,n
III. LASING of the two-levelsystem are faster than the time scales of
the oscillator we can form a closed set of equations
To find the density matrix in the stationary limit, we
describe the system by a master equation. The master d ρ↑,n−1 (38)
equation reads dt(cid:18) ρ↓,n (cid:19)
Γ Γ Γs Γs ρ
1 = − +− −− +,n−1 −,n ↑,n−1
ρ˙ = −i (cid:20)2∆Eσz +Ωa†a,ρ(cid:21) (34) (cid:18) Γs+,n−1 −Γ+−Γ−−Γs−,n (cid:19)(cid:18) ρ↓,n (cid:19)
Γ ρ
+(L +L +L )ρ, + + n−1 .
g ch diss (cid:18) Γ−ρn (cid:19)
wheretheLindbladoperatorisseparatedintotheeparts.
Thissetofequationscanbesolvedinthestationarylimit
The coupling between oscillator and two level system is
andwegetanequationfortheeffectoftheartificialatom
described by
on the oscillator
L ρ = Γs (2o σ ρσ o
g +,n n+1,n − + n,n+1 ρ˙ = γ+ρ γ+ +γ−+κn ρ (39)
Xn n n n−1− n+1 n n
σ o σ ρ ρσ o σ ) + γ− +(cid:0)κ(n+1) ρ , (cid:1)
− + n,n − − + n,n − n+1 n+1
+ Γs (2σ o ρo σ (cid:0) (cid:1)
−,n + n−1,n n,n−1 − with
Xn
−σ−on,nσ+ρ−ρσ−on,nσ+) , γ+ = Γ+Γs+,n−1
n Γ +Γ +Γs +Γs
where we introduced the operator o = n m. Exci- + − +,n−1 −,n
n,m | ih | Γ Γs
tationandrelaxationofthetwolevelsystemiscontained γ− = − −,n .
in the superoperator Lch, which reads n Γ++Γ−+Γs+,n−1+Γs−,n
(40)
Γ
+
L ρ= (2σ ρσ σ σ ρ ρσ σ )
ch + − − + − +
2 − −
It is now relatively simple to write down the solution for
Γ
+ − (2σ ρσ σ σ ρ ρσ σ ). (35) the stationary density matrix, which is given by
− + + − + −
2 − −
The excitation and relaxation rates are given by Γ± = ρ =ρ n γm+ (41)
Γ0± +Γd∓e,cna→y±,n. Here, the rates Γ0± model an external n 0mY=1γm−+κm
6
with the normalization factor ρ . In the case of lasing, B. Large photon numbers
0
the distribution function is peaked around the average
photon number a†a = n . In this section we will calculate the photon number in
h i h i
We can derive a good approximation for the average the limit Γ Γ +Γ . In this case we can write
T + −
≫
photon number using
Γ Γs
γ+ = + +,n−1
γh+ni n Γs+,n−1+Γs−,n
γh−ni+κhni ≈1. (42) γn− = Γs Γ−Γ+s−,Γns
+,n−1 −,n
This condition can be solved in two regimes which we (47)
will discuss next. These two regions are mostly defined
and we can directly find an effective equation for the
by the relation between the photon creation/absorption
number of photons from Eq. (42)
ratesΓ andthebroadeningΓ,whichweintroducedin
±,n
Eq. (23). Thebroadeningcanbecausedbymanyeffects,
Γ +κ n
like the inherent width of the spectral peak around ωL. Γs+,hni−1 = Γ− κhniΓ−,hni . (48)
However, we also get an additional broadening from e.g. +− h i
the pumping Γ [22]. Therefore, we can always assume To find a more explicit result, we need to simplify the
±
Γ&(Γ +Γ )/2. photon creation rates. To do this, we assume we are at
+ −
small temperatures, ω k T, which means η 0,
L B +
≫ ≈
that the resonance condition δω =ω is met, and Γ
L T
≪
ω . Using these conditions we find
A. Small photon numbers L
2Γ3 eη− 1
Γs 2Γ (49)
We willnowconsidera regimewherethe photonnum- +,n−1 ≈ − g2sin2θ η− n
ber remains relatively small, which means that ΓT 2g4sin4θ
Γ++Γ−. We use the connection between the broaden≪ed Γs−,n ≈ Γω2 f(η−)n2
spectralfunctionandthepumpingratesandassumethat L
thebroadeningismostlytheresultofthepumping. This ηn′ 1
f(η ) = η e−2η− − .
gives us Γ+ = Γ− = Γ. To be more precise, we also − − Xn′ n′! (n′+1)2
assume that the resonance condition δω = ω is exactly
L
fulfilled. At low temperatures this allows us to use the For the photon creation rates, we want the first term to
following approximations dominate, which gives us a condition for the coupling
strength g,
1 g4n2S2(ω )sin4(θ)
γ+ = g2nsin2(θ)S (ω ) P L Γ2 eη− 1
n 2 P L − 4Γ 1. (50)
γ− = 0 . (43) g2sin2θ η− hni ≪
n
This means that in principle we would like the effective
Now it is relatively simple to solve the condition coupling between the two level system gsinθ n and
h i
γ+ /κ n = 1. From this we get the average photon the cavity to be similar to the width of the dpissipative
hni h i environment Γ. If the condition (50) is valid we find,
number
Γ2ω2 Γ +κ n
2Γ 4Γκ L = − h i n 2. (51)
hni≈ g2sin2θSP(ωL) − g4sin2θSP(ωL). (44) g4sin4θf(η−) Γ+−κhnih i
For this to have a solution with a large photon number
We can further approximate this solution by assuming n we of course need Γ κ. Additonally it seems
+
h i ≫
that only the resonant peak contributes to S (ω ), beneficial if g ω . As we have seen in condition (50),
P L L
≪
the coupling g should not be too small. Yet at the same
SP(ωL)≈ 4ǫCΓcωoLs2θe−4ǫCωcLos2θ, (45) ctirmeaet,ioωnLainsdrealbevsoarnpttifoonr tahnedatshyemremfoerterylarbgeetwωeLenispahnotimon-
portant ingredient.
which gives us
Γ2ωL 4ǫCcos2θ IV. CONCLUSION
n e ωL (46)
h i ≈ 2ǫ g2sin2θcos2θ
C
We have discussed lasing in the model of a two-level
1 ΓωLκ e4ǫCωcLos2θ . system coupled to a lasing cavity and a dissipative en-
×(cid:18) − 2g2ǫCsin2θcos2θ (cid:19) vironment. The dissipative environment has a spectral
7
density with a maximum at the frequency ω , while k T ω even at room temperatures, and therefore
L B L
≪
the lasing cavity has the frequency Ω. Lasing can be it would allow for a room temperature laser in the THz
generated if we are close to the resonance condition regime.
ω +Ω = ∆E, where ∆E is the energy splitting of the
L
two level system. We see that this allows for many com-
binationsofω andΩandthereforeitallowsforlasingat
L
wide range of frequencies. An important feature is that,
Acknowledgements
while we operate at small temperatures as compared to
thecharacterisitcenergiesofthedissipativeenvironment,
k T ω , the frequency Ω can be much smaller than The study was supported by a research fellowship
B L
≪
temperature. within the project Enhancing Educational Potential
An interesting regime where these results could be of Nicolaus Copernicus University in the Disciplines
relevant, is for quantum dots in the optical domain of Mathematical and Natural Sciences (project no.
with a cavity at THz-frequencies. This would allow for POKL.04.01.01-00-081/10.).
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