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Controlling the emission and absorption spectrum of a quantum emitterin a dynamic environmentTo cite this article: H F Fotso 2019 J. Phys. B: At. Mol. Opt. Phys. 52 025501
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Controlling the emission and absorptionspectrum of a quantum emitter in a dynamicenvironment
H F Fotso
Department of Physics, University at Albany (SUNY), Albany, NY 12222, United States of America
E-mail: [email protected]
Received 19 June 2018, revised 9 October 2018Accepted for publication 27 November 2018Published 14 December 2018
AbstractWe study the emission spectrum and absorption spectrum of a quantum emitter when it is drivenby various pulse sequences. We consider the Uhrig sequence of nonequidistant πx pulses, theperiodic sequence of πxπy pulses and the periodic sequence of πz pulses (phase kicks). We findthat, similar to the periodic sequence of πx pulses, the Uhrig sequence of πx pulses has emissionand absorption that are, with small variations, analogous to those of the resonance fluorescencespectrum. In addition, while the periodic sequence of πz pulses produces a spectrum that isdependent on the detuning between the emitter and the pulse carrier frequency, the Uhrigsequence of nonequidistant πx pulses and the periodic sequence of πxπy pulses have spectra withlittle dependence on the detuning as long as it stays moderate along with the number of pulses.This implies that they can also, similar to the previously studied periodic sequence of πx pulses,be used to tune the emission or absorption of quantum emitters to specific frequencies, tomitigate inhomogeneous broadening and to enhance the production of indistinguishable photonsfrom emitters in the solid state.
Keywords: emission/absorption spectrum, spectral diffusion, pulse-driven emitter, solid stateemitter
(Some figures may appear in colour only in the online journal)
1. Introduction
Understanding the dynamics of a quantum emitter coupled toan electromagnetic field, in addition to being a longstandingfundamental question [1, 2], is also of significant currenttechnological interest. In particular, the ability to control theemission or absorption spectrum of a quantum emitter will beof great value to the fields of quantum control, quantummetrology or quantum information processing (QIP) [3–6].
Indeed, many promising quantum systems that are primecandidates to serve as quantum bits (or qubits) in QIP arequantum emitters in the solid state [7–20]. As such, theiremission and absorption spectra are subject to fluctuations intheir environment that can lead to the spectrum drifting ran-domly in time: spectral diffusion [21, 22]. This phenomenonreduces the efficiency of fundamental operations in QIP suchas the photon-mediated entanglement of distant quantum
nodes or the coupling of quantum nodes to photonic cavities.It has for this reason received a great deal of attention [10–14,21, 23–33].
In addition, the absorption spectrum of a quantum emittersuch as a Nitrogen-vacancy (NV) center in diamond can beused to probe weak electromagnetic fields, temperatures orforces with very high spatial resolutions. Because of inho-mogeneous broadening from the ensemble of emitters in theprobe, the sensitivity is limited by T1 2* instead of 1/T2[34–36].
Recent work has explored the possibility of using ade-quate pulse sequences to control the spectrum of a quantumemitter [23, 37, 38]. In particular, it was shown that a periodicsequence of πx pulses can be used to suppress spectral dif-fusion from photons produced by a quantum emitter in adiffusion-inducing environment thus enhancing photonindistinguishability [23]. The absorption spectrum of quantum
Journal of Physics B: Atomic, Molecular and Optical Physics
J. Phys. B: At. Mol. Opt. Phys. 52 (2019) 025501 (8pp) https://doi.org/10.1088/1361-6455/aaf452
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emitters subjected to the same pulse sequence was also stu-died theoretically [39] and experimentally shown to enhancethe sensitivity of NV centers used in magnetometry to the1/T2 limit [40].
These early studies focused on a periodic sequence of πxpulses. The underlying mechanism for modifying the spectrallineshapes is clearly different from that of coherence protec-tion with dynamical decoupling protocols. In the context ofdynamical decoupling, the objective is to decouple the emitterfrom the decoherence inducing bath. In the current problem,our goal is not to decouple the emitter from the radiation bath,but rather to ensure emission/absorption at specific fre-quencies independent of the environment. However, in lightof the similarities with dynamical decoupling where differentpulse sequences are used with widely varying degrees ofsuccess, it is natural to inquire about the fate of the emissionand absorption spectra of quantum emitters if they are drivenby other pulse sequences. In this paper, we analyze theemission and absorption spectra of quantum emitters whenthey are driven by different other pulse sequences. We studythe Uhrig sequence [41] of nonequidistant πx pulses, theperiodic sequence of πx πy pulses and the periodic sequenceof πz pulses.
We find that the Uhrig sequence of πx pulses has emis-sion and absorption spectra that are analogous to those of theresonance fluorescence spectrum. In particular, the emissionspectrum, has a lineshape analogous to the Mollow triplet of aresonantly driven two-level system (TLS) [42–44] with mostof the spectral weight at a central peak flanked with twosatellite peaks; similarly, the absorption spectrum is similar tothat of a TLS continuously driven on resonance and displays alocal maximum at the pulse carrier frequency unlike that ofthe periodic sequence of πx pulses that has a local minimum atthe pulse carrier frequency. The periodic sequence of πxπypulses has emission and absorption spectra that are qualita-tively different from those of πx pulse protocols. They havetheir main peak at π/2τ and satellite peaks at −π/2τ and π/2τ+π/τ, where τ is the time interval between successivepulses. The periodic sequence of πz pulses has a similarlineshape in both its emission and its absorption spectrumwith two peaks of equal spectral weight at Δ−π/τ andΔ+π/τ. Where Δ is the detuning between the emitter andthe pulse carrier frequency. The Uhrig sequence of πx pulsesand the periodic sequence of πxπy pulses have spectra withlittle dependence on the detuning of the emitter with respectto the pulse carrier frequency as long as it stays moderatealong with the number of pulses. This implies that they canalso be used to tailor the emission or absorption of quantumemitters, to mitigate inhomogeneous broadening and toenhance the production of indistinguishable photons fromemitters in dynamic environments.
The rest of the paper is organized as follows. In section 2,we describe the model of the quantum emitter coupled to theradiation field and controlled by various pulse sequences, andthe master equations governing the system dynamics. Insection 3, we describe the methods used to obtain the emis-sion and absorption spectra. In section 4 we present resultsthat show the ability to tune the emission and absorption
spectra with appropriate pulse sequences. Section 5 presentsour conclusions.
2. Modeling the driven emitter + radiation system
The quantum emitter can be modeled as a TLS with groundstate gñ∣ and excited state eñ∣ , separated in energy byE Ee g 1 0 w w- = = + D( ) (figure 1(a)). Below we setÿ=1. The TLS is coupled to normal modes of the electro-magnetic radiation field, and is, at appropriate times, drivenby pulses of the laser field with the Rabi frequency Ω. Initi-ally, at time t=0, the excited state is assumed to be occupiedand the ground state to be empty; additionally, all bosonicmodes are initially assumed to be empty. The Hamiltoniandescribing this system can be written as [45]:
H a a i g a a
d E
2
. 1
zk
k k kk
k k k
e
1 å åws w s s= + - -
-
- +
( )
· ( )
† †
The first term corresponds to the TLS, the second term to theradiation bath, the third term to the coupling between the TLSand the radiation bath written in the rotating-wave approx-imation (RWA). The last term corresponds to the coupling ofthe TLS with the external driving field. d
is the electric dipole
moment of the TLS and Ee
is the external driving field that is
applied at times prescribed by the pulse sequence. It hasamplitude E
such that the Rabi frequency is di iEW =
· . We
have introduced the standard pseudo-spin Pauli operators forthe TLS: e e g gzs = ñá - ñá∣ ∣ ∣ ∣, e gs = ñá+ ∣ ∣ and
Figure 1. (a) Schematic representation of a two-level system withground state gñ∣ and excited state eñ∣ separated by energyω1=ω0+Δ. (b) In the absence of any driving field, the emissionand absorption spectra have a Lorentzian lineshape centered aroundω1. (c) Uhrig Sequence of πx pulses. (d) Periodic sequence of πxπypulses. (e) Periodic sequence of πz pulses.
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J. Phys. B: At. Mol. Opt. Phys. 52 (2019) 025501
g es s= ñá =- +∣ ∣ ( )†. ak† and ak are respectively the creation
and the annihilation operator for a photon of mode k with thefrequency ωk, and gk is the coupling strength for this mode tothe TLS.
For a given driving sequence, the last term of (1) can beexpanded and written in the RWA. For πx rotations, therelevant Hamiltonian is thus, in the frame rotating at fre-quency ω0:
H a a i g a a
t
2
2,
2kk k k z
kk k k
x
å åw s s s
s s
= +D
- -
+W
+
- +
+ -
( )
( ) ( )( )
† †
where all energies are now measured with respect to ω0 and
1 0w wD = - is the detuning of the TLS’s transition fre-quency from the pulse carrier frequency. The time-depend-ence Ωx(t) is determined by the pulse sequence. We considerpulses such that Ωi(t)=Ωi during the pulses and zerootherwise. We will assume Ωi to be much larger than all otherrelevant energy scales so that the pulses are essentiallyinstantaneous (i.e. g, ,i kW D G ). Previous studies haveshown that imperfect or finite width pulses do not sig-nificantly change the spectral lineshapes [23]. While the lastterm in (1), written in the form of (2), amounts to treating theincident field as a classical time-dependent field, it can beshown to be equivalent to the treatment of an initiallycoherent incident field via a unitary transformation[42, 45, 46].
In the absence of all control (Ωi(t)=0 for all times), thesystem exhibits spontaneous decay, and the correspondingemission rate is g dk2 ;k k
2òp d wG = - D( ) We normalizeour energy and time units so that Γ=2, and the corresp-onding spontaneous emission line has a simple Lorentzianshape 1 12w +( ), with the half-width equal to 1. This way,all frequencies are measured in units of Γ/2.
To characterize the dynamics of the system and obtainthe emission and absorption spectrum, we analyze the time-evolution of the density matrix of the emitter, which is writtenas
t t e e t e g
t g e t g g , 3ee eg
ge gg
r r r
r r
= ñá + ñá
+ ñá + ñá
( ) ( )∣ ∣ ( )∣ ∣( )∣ ∣ ( )∣ ∣ ( )
with ge eg*r r= . The master equations governing the time-
evolution of the density matrix operator can be obtained fromthe Hamiltonian (2) by using the approximation of indepen-dent rate of variations whereby we independently add to thetime-evolution of each matrix element of ρ, terms due to theradiation bath, the incident field and the damping termsresponsible for spontaneous emission [45]. For πx pulses andfor the TLS described by the above Hamiltonian (2), the
master equations in the rotating-wave approximation are:
it
it
i it
i it
2,
2,
2 2,
2 2. 4
eex
eg ge ee
ggx
eg ge ee
ge gex
ee gg
eg egx
ee gg
r r r r
r r r r
r r r r
r r r r
=W
- - G
=-W
- + G
= D -G
-W
-
= - D -G
+W
-
⎜ ⎟
⎜ ⎟
⎛⎝
⎞⎠
⎛⎝
⎞⎠
˙ ( ) ( )
˙ ( ) ( )
˙ ( ) ( )
˙ ( ) ( ) ( )
Each πx pulse inverts the populations of the excited andground state and swaps the values of ρeg and ρge.
Similarly, each πy pulse inverts the populations of theexcited and ground states and swaps the values of ρeg with−ρge and vice versa. We will also consider πz pulses that areequivalent to π phase kicks leaving ρee and ρgg unchangedand replacing ρeg by −ρeg and ρge by −ρge. In general, theeffect of a πi pulse can be summarized as:
0 0 , 5ni
nir s r s=+ -( ) ( ) ( )( ) ( )
where 0nr -( )( ) and 0nr +( )( ) are the density matrices imme-diately before and immediately after the n th pulse. σi is thepseudo-spin Pauli matrix in the i-direction in which the pulseapplies the rotation.
The emission spectrum can be obtained using a narrow-band detector that can be modeled as a two-level absorberwith a very sharp transition frequency [47]. The excitationprobability of this detector then corresponds to the emissionspectrum. At long times T, it can be expressed as:
P A
dt d t t i
2
Re exp .
6
T T t
2
0 0ò ò
w
q s q s wq
=
´ á + ñ --
+ -⎧⎨⎩
⎫⎬⎭
( )
( ) ( ) [ ]( )
ts-( ) and ts q++( ) are the time-dependent operators in theHeisenberg representation, and the angled brackets representthe expectation values that are taken with respect to the initialstate of the TLS (in our case, fully occupied excited state andempty ground state). The constant A is independent of thepulse parameters, does not affect the spectral shape and onlyaffects the absolute scale of the spectrum.
On the other hand, the absorption spectrum consideredhere is measured by determining the energy absorbed from aweak probing field by the TLS while it is simultaneouslybeing driven by the relevant pulse sequence. Since theprobing field is assumed to be weak enough that its effects onthe populations of the states can be neglected, the absorptionspectrum can be calculated within the linear response theory.The absorption as a function of frequency, Q(ω), is given by[48, 49]
Q A
dt d t t
2
Re , e ,
7
T T ti
2
0 0ò ò
w
q s s q
=
´ á + ñ wq-
- +-
⎧⎨⎩⎫⎬⎭
( )
[ ( ) ( )]( )
[O1, O2] is the commutator of the operators O1 and O2. Forthe absorption spectrum, the expectation value is evaluated in
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J. Phys. B: At. Mol. Opt. Phys. 52 (2019) 025501
the absence of the probing field. The expression (7) can berewritten as
Q A
P P
2 Re, 8
22 1 w w w
w w= -= ¢ -
( ) { ( ) ( )}( ) ( ) ( )
where
dt d t t e 9T T t
i2
0 0 ò òw q s s q= á + ñ wq
-
- +-( ) ( ) ( ) ( )
and
dt d t t e . 10T T t
i1
0 0 ò òw q s q s= á + ñ wq
-
+ --( ) ( ) ( ) ( )
The term P A2 Re21w w=( ) { ( )} can be recognized to be the
emission spectrum of equation (6). P A2 Re22w w¢ =( ) { ( )}
can be viewed as the direct absorption so that the differenceyields the net absorption [42]. To obtain the absorptionspectrum, both terms are evaluated independently before thentaking the difference to obtain Q(ω).
In the absence of any pulses, as illustrated in figure 1(b),the emission spectrum and absorption spectrum have Lor-entzian-shaped profiles centered at the emitter’s frequencythat is equal to the detuning Δ (in the frame rotating at ω0).
The two-time correlation function t ts q sá + ñ+ -( ) ( ) thatappears in the expression for the emission spectrum P(ω) isusually expressed as a single-time expectation value[2, 42, 47] following:
t t
U t U t U t U tTr 0 0, 0, 0, 0,
11
s q s
r q s q s
á + ñ
= + +
+ -
+ -
( ) ( )
[ ( ) ( ) ( ) ( ) ( )]( )
† †
t U t t U t tTr , , 12s r q s q= + +- +[ ( ) ( ) ( )] ( )†
tTr 13r q s= ¢ + +[ ( ) ] ( )
where t tr s r¢ = -( ) ( ), and where σ+ and σ−are the time-independent operators in the Schrödinger picture. U t t, ¢( ) isthe time-evolution operator for the system described byequations (4). Similarly, to evaluate P w¢( ), we rewrite theinvolved two-time correlation function as:
t t tTr 14s s q s r qá + ñ = +- + +( ) ( ) [ ( )] ( )
with t tr r s = -( ) ( ) .
3. Analytical and numerical solutions
The expectation values (13) and (14), and the emission andabsorption spectra, can be evaluated numerically, or forsimple pulse sequences, analytically. Both methods, asdemonstrated previously [39], are in perfect agreement.
While the analytical solution of the master equation is ingeneral rather tedious, for a periodic sequence of instanta-neous pulses, one obtains a set of decoupled first order diff-erential equations that can be solved in a procedure similar tothat of [39] where the obtained analytical solution was alsoshown to overlap with the numerical solution. Alternatively,one can also, in this case of periodic instantaneous pulses,integrate the Heisenberg equations of motion in a toggling
frame. We outline this solution here for the calculation of theemission spectrum of a quantum emitter driven by a periodicsequence of πxπy pulses. In this situation, the Hamiltonianafter the nth pulse can be rewritten as:
H t a a
i g a a a a
12
,
15
nz
kk k k
kk k k k k1 1 2 2
å
å
s w
c s c s c s c s
= -D
+
+ - + -- + + -
( ) ( )
{ }( )
†
† †
where t t t1 1 3c x x= -( ) ( ) ( ) and t t t2 2 4c x x= -( ) ( ) ( ) withξ1,2,3,4 periodic functions defined in the interval [0, 4τ] by:
t t tfor 0 ; 1, 0, 161 2,3,4t x x< < = =( ) ( ) ( )
t t tfor 2 ; 1, 0, 172 1,3,4t t x x< < = =( ) ( ) ( )
t t tfor 2 3 ; 1, 0, 183 1,2,4t t x x< < = =( ) ( ) ( )
t t tfor 3 4 ; 1, 0. 194 1,2,3t t x x< < = =( ) ( ) ( )
From the above Hamiltonian, one obtains after the nth pulsethe equations of motion:
a i a g 20k k k k 1 2w c s c s= - + +- +˙ ( ) ( )
i g a g a1 21n
kk k z
kk k z1 2å ås s c s c s= - - D + -- -˙ ( ) ( )†
g a a a a2 .
22
zk
k k k k k1 1 2 2ås c s c s c s c s= - + - -- + + -˙ [ ]
( )
† †
Starting from an initial configuration at t=0 in which allbosonic modes are empty and the emitter has its excited statefully occupied and its ground state empty, we want to eval-uate the number N a t a tk k k= á ñ( ) ( )† of bosons in bosonicmode k at time t Nt d= + where N is a multiple of 4 and
0,d tÎ [ ]. To this end, we will recursively integrate theabove equations of motion for the annihilation operator akafter successive pulses. After N pulses, employing the Mar-kovian approximation, we will get:
a t a g t dt
g t t dt
g t t dt
0 e e
e
e . 23
kN
ki t
kN
tN i t t
kl
Nl i t t l
kl
Nl i t t l
1 1
0
4 1
0
4
1 2 24
2
0
4 1
0
4
2 2 24
2
k k
k
k
1
2
2
ò
ò
ò
å
å
s
c s
c s
= +
+
+
w
t
w
tw t
tw t
--
- -
=
-
-- - -
=
-
+- - -
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
To be able to utilize this expression, we now need to obtain,s- + after each pulse interval. Within the Markovian
approximation, integrating for s t-( ) gives:
e 0 24i i 0s t s= t g t-
- D +-( ) ( ) ( )( ( ))( )
with g gk k i k k0 2 2 e 1
k
i k
k2g t = å - åt
w w-D-
-D
w t- -D( )( )
( ) ( )
( ). Similarly,
integrating for 2s t-( ) will give:
2 e 25i i 0 *s t s t= t g-
D +-( ) ( ) ( )( )( )
e e 0 26i i i i0 0* s= t g t gD + - D +-( ) ( )( ) ( )( ) ( )
e 0 . 27i2 Im 0 s= g-( ) ( )( )( )
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J. Phys. B: At. Mol. Opt. Phys. 52 (2019) 025501
Here, one immediately notices the cancellation of Δ
between (26) and (27). This is the aforementioned phasecancellation that is responsible for the spectrum not depend-ing on the detuning with respect to the pulse-carrier fre-quency. This cancellation is due to the change of sign in thefirst term of the Hamiltonian of equation (15) between con-secutive pulses.
By proceeding in this iterative integration, one obtains,for an integer n<N/2:
n2 e 0 , 28i n2 Im 0s t s= g- -( ) ( ) ( ){ }( )
n2 1 e 0 . 29i n i2 Im 0 0s t s+ = g g t-
+ - D-(( ) ) ( ) ( ){ }( ) ( )
We also get after the Nth pulse:
t 0 e e 30N iN i t N i t NIm 0s s= g t t f t- -
- D - + -( ) ( ) ( )( ) ( ( )) ( ( ) ( ))( )
with t gk kt N
i2 e 1
k
i k t N
k2f = å -t
w w--D
--D
w t- -D -⎡⎣ ⎤⎦( )( ) ( )
( )( ).
These expressions, along with the correspondingexpressions for σ+ can then be plugged into equation (23)allowing for the evaluation of the emission spectrum in termsof expectation values at time t=0. Making use of
I 2zs s s= -- + ( ) and I 2zs s s= ++ - ( ) and of the van-ishing of expectation values of s s- - and s s+ +, the nonzeroterms can be collected into a straightforward but lengthyexpression that will not be displayed here. These nonzeroterms can then be evaluated numerically and summed up toproduce the spectrum. The resulting spectrum has a lineshapethat is, with respect to peak positions and relative spectralweights, consistent with that obtained by either analytical ornumerical solution of the master equation.
For the numerical solution of the master equation (4), thetime axis is divided in finite pulse intervals separated byconsecutive pulses, and each pulse interval is discretized insmaller steps of length Δt. Starting at t=0, with the knowninitial conditions, 1, 0, 0, 0ee gg eg ger r r r= = = = , weintegrate equation (4) to evolve the matrix elements
, , ,ee eg ge ggr r r r from time t to t+Δt and iterate this inte-gration up to the first pulse time T1. We then apply the pulseto the system (i.e equation (5) to the density matrix operator)before resuming the iterative integration starting from time T1and up to T2, the time at which the next pulse is applied. Thisprocess is repeated until time T TNp = where Np is the totalnumber of pulses. It allows us to obtain, tr¢( ) and ρ″(t) fort T0,Î [ ]. We can then proceed, once again, with the inte-gration of equation (4) starting at t T0,Î [ ] to obtain
tr q¢ +( ) and tr q +( ) for T t0,q Î -[ ]. It is thereonstraightforward to obtain t ts q sá + ñ+ -( ) ( ) and
t ts s qá + ñ- +( ) ( ) from equations(13) and (14) respectively.We finally get the emission spectrum and the absorptionspectrum by performing the relevant Fourier transforms withrespect to θ and integration over t to get 1 w( ) and 2 w( ). Thereal part of 1 w( ) gives the emission spectrum and the dif-ference between the real parts of 2 w( ) and 1 w( ) gives theabsorption spectrum. The results that we present next areobtained using this numerical approach that treats all pulsesequences on the same footing.
4. Results
4.1. Uhrig pulse sequence
The Uhrig pulse sequence was recently suggested as asequence of nonequidistant pulses that is optimal in pre-venting decoherence due to low frequency noise in theenvironment [41]. A cycle of the Nth order Uhrig pulsesequence is made of N pulses applied at times given by:
T Tj
Nj Nsin
2 1, with 1, 2, , . 31j
2 p=
+=
( )( )
The (N+1)th pulse is applied at TN+1=T and the timesequence is repeated. The Uhrig pulse sequence is illustratedin figure 1(c). Considering the importance of this pulsesequence in the dynamical decoupling context, it is is usefulto study how it affects the emission or absorption spectrum ofa quantum emitter in a diffusion-inducing bath. In figure 2,we present results for a quantum emitter driven by a Uhrigsequence of πx pulses. The emission spectrum is found tohave a central peak at the pulse carrier frequency, flanked bytwo satellite peaks the position of which is dependent on thenumber of pulses. The emission and absorption spectra do notdepend much on the detuning Δ as long as it remains mod-erate (figures 2(a)–(c)). For the same detuning Δ and fixed
Figure 2. Emission and absorption spectra of TLS driven by a Uhrigsequence of πx pulses. (a) Emission for different values of thedetuning Δ between the transition frequency and the pulse carrierfrequency (Δ=3.0 (red), Δ=4.0 (green), Δ=5.0 (magenta),Δ=6.0 (blue)) after 12 pulses applied during time T=2.0. (b)Emission for fixed detuning (Δ=3.0) with 6 pulses (doted magentaline), 8 pulses (dashed blue line), 10 pulses (dashed green line), 12pulses (red line) after time T=2.0. (c) Emission for fixed detuning(Δ=3.0) with 12 pulses applied during a total time T=1.0 (redcircles), T=1.2 (green diamonds), T=1.6 (magenta stars),T=2.0 (blue triangles). (d) Absorption for fixed detuning(Δ=3.0) with 12 pulses applied during a total time T=1.0 (redcircles), T=1.2 (green diamonds), T=1.6 (magenta stars),T=2.0 (blue triangles).
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J. Phys. B: At. Mol. Opt. Phys. 52 (2019) 025501
time T, the central peak is of similar spectral weight andsatellite peaks move further away from it as the number ofpulses increases (figure 2(b)). We note a strong analogy withthe spectrum of the resonance fluorescence problem. Theabsorption spectrum has both positive and negative parts withthe former often interpreted as true absorption and the latter asstimulated emission in the direction of the probing field[48, 50, 51]. Markedly, the absorption spectrum has a localmaximum at the pulse carrier frequency (figure 2(d)) unlikethat of a periodic sequence of πx pulses that has a localminimum at the pulse carrier frequency.
4.2. πx πy pulse sequence
In figure 3, we present the results for an emitter driven by aperiodic sequence of πxπy pulses. The emission spectrum hasits main peak located at π/2τ with additional peaks at −π/2τand π/2τ+π/τ (figure 3(c)) unlike the periodic sequence ofπx pulses for which the main peak is at the pulse carrierfrequency (ω=0 in the frame rotating at the pulse carrierfrequency). For fixed detuning, the τ-dependence of theemission spectrum is shown in figure 3(b). The lineshape isestablished early and the peaks grow in amplitude with time(figure 3(c)). The absorption spectrum has its main dip at π/2τ and two additional dips at −π/2τ and π/2τ+π/τ
(figure 3(d)). Both the emission and the absorption spectrashow little dependence on Δ for moderate values of the pulsespacing time τ as long as Δ 1/τ (figure 3(a)).
4.3. πz pulse sequence
Contrary to the πx and the πxπy pulse sequences, a sequenceof πz pulses (or π phase kicks), produces a spectrum thatdepends on the detuning between the pulse carrier frequencyand the TLS frequency (4(a)). This can be related to the factthat the toggling frame Hamiltnian in this case does not dis-play a sign change between consecutive pulses such as that ofπx or πy pulses that is seen in equation (15). Both theabsorption spectrum and the emission spectrum have similarlineshapes. Thus we only present in figure 4 the emissionspectrum for πz pulses. The emission spectrum is split in twopeaks of equal weight located at Δ−π/τ and Δ+π/τ(4(b)–(c)). Thus, for the same pulse sequence, it dependsstrongly on the values of Δ (figure 4(a)). These results are inagreement with those of [37] where a model of the atom +radiation system was solved using direct diagonalization for afinite but large number of bosonic modes.
5. Conclusions
We have studied the emission and absorption spectra of aquantum emitter when it is driven by a pulse sequence.
Figure 3. Emission and absorption spectra of TLS driven by aperiodic sequence of πx πy pulses. (a) Emission for different valuesof the detuning Δ between the transition frequency and the pulsecarrier frequency (Δ=3.0 (red), Δ=4.0 (green), Δ=5.0(magenta), Δ=6.0 (blue)) after 12 pulses with a time spacingτ=0.2 between successive pulses. (b) Emission for fixed detuning(Δ=3.0) with 6 pulses and τ=0.4 (red line), 8 pulses and τ=0.3(dashed blue line), 12 pulses and τ=0.2 (dashed green line). (c)Emission for fixed detuning (Δ=3.0) and τ=0.2 after 6 pulses(red circles), 8 pulses (green diamonds), 10 pulses (magenta stars),12 pulses (blue triangles). (d) Absorption for fixed detuning(Δ=3.0) after 6 pulses (red circles), 8 pulses (green diamonds), 10pulses (magenta stars), 12 pulses (blue triangles).
Figure 4. Emission spectrum of TLS driven by a periodic sequenceof πz pulses (phase kicks). (a) For different values of the detuning Δbetween the transition frequency and the pulse carrier frequency(Δ=3.0 (red), Δ=4.0 (green), Δ=5.0 (magenta), Δ=6.0(blue)) after 12 pulses with a time spacing τ=0.2 betweensuccessive pulses. (b) For fixed detuning (Δ=3.0) with 6 pulsesand τ=0.4 (doted magenta line), 8 pulses and τ=0.3 (dashed blueline), 12 pulses and τ=0.2 (dashed green line), 24 pulses andτ=0.1 (red line). (c) For fixed detuning (Δ=3.0) and τ=0.2after 6 pulses (red circles), 8 pulses (green diamonds), 10 pulses(magenta stars), 12 pulses (blue triangles).
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Representing the quantum emitter as a TLS, we have used themaster equation governing the time-evolution of the densitymatrix operator of the pulse-driven system to obtain thespectrum for the different protocols. We have considered thecase of the Uhrig sequence of πx pulses, a periodic sequenceof πx πy pulses and a periodic sequence of πz phase kicks. Inthe absence of any driving protocol, the emission andabsorption spectra have Lorentzian lineshapes centeredaround the frequency Δ=Ee−Eg (measured in the framerotating at the target frequency ω0). The periodic sequence ofπz phase kicks splits the emission spectrum in two peaks ofequal weight at Δ+π/τ and Δ−π/τ. It also modifies theabsorption spectrum in a similar manner. The Uhrig sequenceof πx pulses has an emission and absorption spectrums similarto that of a periodic sequence of πx pulses where a centralpeak with the bulk of the emission/absorption appears at thepulse carrier frequency with satellite peaks at positive andnegative frequencies dependent on the number of pulses. Inaddition, its absorption spectrum displays a strong analogywith that of the resonance fluorescence with a peak (a max-imum) at the pulse carrier frequency as opposed to theminimum that is observed for the periodic sequence of πxpulses. The periodic sequence of πx πy pulses produces anemission spectrum with the main peak located at π/2τ andsatellite peaks at −π/2τ and π/2τ+π/τ. Similarly, for theUhrig sequence of πx pulses, the spectra show little depend-ence on the detuning for a moderate number of pulses overthe emission time. These results provide a detailed picture forthe emission and absorption spectra of a quantum emitterwhen it is driven by different pulse sequences. They alsoindicate that the Uhrig pulse sequence and the periodicsequence of πxπy pulses can be used to control the effect ofthe environment on the emission and absorption spectrum. Ashighlighted in the analytical solution, the phase cancellationresponsible for the limited dependence of the spectrum on theenvironment will occur as long as the detuning with respect tothe pulse carrier frequency and the time between consecutivepulses remain moderate (for periodic pulses: Δ·τ∼1).Most importantly, this phase cancellation argument is madewith no additional regard to the underlying mechanismresponsible for the fluctuations in Δ. This suggests that theseresults can be observed in a variety on quantum emitters. Forinstance, 3 to 4 pulses per free emission time should sufficefor NV centers in diamond.
Acknowledgments
We thank V V Dobrovitski for inspiring this work and forinsightful discussions. We thank V Mkhitaryan for helpfuldiscussions.
ORCID iDs
H F Fotso https://orcid.org/0000-0001-7952-6256
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