PHYSICAL REVIEW A
VOLUME 59, NUMBER 5
MAY 1999
Polychromatic excitation of a two-level system
Andrew D. Greentree, Changjiang Wei, and Neil B. Manson
Laser Physics Centre, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, 0200,
Australian Capital Territory, Australia
~Received 2 October 1998!
We report on the Autler-Townes spectra for a two-level atom driven by multiple fields. We investigate
resonance phenomena and explain our results in terms of multiply dressed states. @S1050-2947~99!02405-1#
PACS number~s!: 42.50.Hz, 32.80.Wr
Phenomena associated with a two-level atom driven by a
strong monochromatic field are frequently analyzed in terms
of dressed states @1#. The dressed states are the eigenstates of
the coupled atom plus driving field system. A dressed-state
approach has previously been extended from one field to two
driving fields, that is, a bichromatically driven two level
atom. The doubly dressed states have been shown to be useful in predicting @2# and explaining observed spectral features @3,4#. The purpose of this paper is to show that the
dressed state formalism can be extended to cover the case of
multiple driving fields, the so called polychromatic driven
two-level atom, and the multiple dressed states can provide a
good account of experiments involving several driving fields.
We consider a two-level atom, with ground state u a & and
excited state u b & and a transition frequency v ab . The u a &
2 u b & transition is driven by multiple fields. The individual
driving fields are numbered, with the ith field having frequency v i , detuning D i 5 v i 2 v ab , and Rabi frequency x i .
In the experiment presented here, up to four strong fields
were simultaneously applied and the field configuration for
four fields is shown in Fig. 1. The effect of the pump fields
was monitored by applying a weak field of frequency v p and
detuning D p 5 v p 2 v ac , which probes the transition from
the ground state, u a & , to the third state, u c & , that is, the
Autler-Townes configuration @5#.
Observed resonances of the system correspond to the
eigenstates of the Hamiltonian of the atom coupled to
~dressed by! the driving fields. In general, the eigenstates of
the Hamiltonian for a polychromatically driven two-level
atom are grouped in an infinite number of manifolds. Because of this complexity, the solutions are most conveniently
obtained by numerical methods. It is observed experimentally that the Autler-Townes spectrum of a polychromatically
driven two-level atom has a complicated multipeak structure.
However, instead of tackling the general case, we present a
special case where we are able to give a simple physical
explanation of the observed spectra. In our specially designed experiment, field i is significantly weaker than field
i21, with the ith field resonant with a transition between the
dressed states associated with the other (i21) fields. The
interaction between atom and field is defined by the operator
x 1 . When the atomic states and the field states are combined,
the resulting atom plus field states, u a & u n 1 & and u b & u n 1 21 & ,
have equal energies. The interaction between the atom and
field mixes these two states into linear combination states @1#
u B,n 1 & 5
1
A2
~ u a & u n 1 & 1 u b & u n 1 21 & ),
~2!
u A,n 1 & 5
1
A2
~ u a & u n 1 & 2 u b & u n 1 21 & ).
The degeneracy is lifted by the atom-field interaction and the
separation between these states is equal to the Rabi frequency of the applied field x 1 ~Fig. 2!. We define the dressed
state separation caused by field i as d i . It is clear that for the
first field d 1 5 x 1 .
We now introduce the second field resonant with one of
the singly dressed state transitions. For example, we can
choose the transition ~i! to be resonant with that between the
dressed states u B,n 1 & and u A,n 1 11 & as shown in Fig. 3.
Strictly, the second field will interact with all four transitions, but in three cases the field is off resonant by x 1 or 2 x 1 ,
and as this detuning is significantly larger than the Rabi frequency x 2 , off-resonant effects can be neglected to first order. We therefore need only consider the resonant interaction
The first field, i51, is applied on resonance ~i.e., D 1
50) with the two-level atom and has a Rabi frequency of
FIG. 1. Energy level and field configuration. The levels are labeled u a & , u b & , and u c & in order of increasing energy. The u a &
2 u b & system forms the two-level system, which is probed by a
weak field frequency v p swept through the u a & 2 u c & transition.
Field 1 is applied on resonance with the u a & 2 u b & transition and the
other fields i have frequency v i , detuning D i , and Rabi frequency
xi .
1050-2947/99/59~5!/4083~4!/$15.00
4083
V5
(i Vi 5 x i~ u a,n i 11 &^ b,n iu 1 u b,n i 21 &^ a,n iu ! .
~1!
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©1999 The American Physical Society
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FIG. 2. Configuration and experimental spectrum for a single
driving field. The bare states are shown on the left and the dressed
states on the right. The resonant field has a frequency of 4.6 MHz
and a Rabi frequency x 1 5120.6 kHz. The lower level is probed in
the neighborhood of a transition at 5.3 MHz and is measured using
the Raman heterodyne detection.
which mixes the degenerate states u B,n 1 & u n 2 & and u A,n 1
11 & u n 2 21 & . To first order, the interaction causes a splitting
given by
d 2 5 u ŠA,n 1 11z^ n 2 21 u Vu B,n 1 & zn 2 ‹u
1
1
5 Šbz^ n 2 21 u V2 u a & zn 2 ‹5 x 2 .
2
2
~3!
Thus the effective interaction strength of the second field
when resonant with a dressed state transition is half that
when resonant with the bare atom transition. The doubly
dressed states, expressed as combinations of the singly
dressed states, are
u B 8 ,n 1 ,n 2 & 5
1
A2
~ u B,n 1 & u n 2 & 1 u A,n 1 11 & u n 2 21 & ),
~4!
u A 8 ,n 1 ,n 2 & 5
1
A2
~ u B,n 1 & u n 2 & 2 u A,n 1 11 & u n 2 21 & ).
Note that all the states dressed by the first field are coupled
by the second field as indicated in Fig. 3. Therefore, the first
field gives rise to a ladder of dressed state doublets and the
second field causes a splitting in each of these doublet levels.
The dressed levels give an allowed transition to the third
state u c & through the transition matrix element
Šaz^ n p u Vp u c & zn p 21‹ with the photon numbers of the driving
fields conserved. Indeed it can be readily seen that the relevant term of the wave function is equally distributed between the four dressed levels and so all four transitions have
equal intensity. The Autler-Townes spectrum is then expected to comprise of a pair of doublets with an intradoublet
separation of d 2 5 x 2 /2 and an interdoublet separation of d 1
5 x 1.
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FIG. 3. Field configuration and spectra in the presence of two
strong pump fields and probed to a third level, with x 1 5120.6 kHz,
D 1 50 kHz. The upper diagram gives the bare states, the single field
dressed states, and the doubly dressed states. Trace ~i! gives the
response when the second field is detuned to the lower frequency
dressed state transition and trace ~ii! corresponds to resonance with
the higher frequency dressed state transition.
The analysis is very similar when the second field couples
the u A,n 1 & state with the u B,n 1 11 & state as indicated by
transition ~ii! in Fig. 3. The strength of the interaction is
again reduced by a factor of 2 and, hence, the four dressed
levels have the same energies as for the previous driving
field. Each level gives an allowed transition to the third
atomic level u c & and so an equivalent four-line AutlerTownes spectrum is expected.
For the three field experiment the first field is resonant
with the two-level transition and the second with the
u B,n 1 & 2 u A,n 1 11 & dressed state transition. With just the
two fields, there will be four doubly dressed levels for each
step of the ladder and 16 transitions between adjacent steps
in the ladder. However, of these transitions there are only
four non-degenerate transitions, indicated by transitions ~i!–
~iv! in Fig. 4. The third field is applied resonant with one of
these transitions and the field further dresses the system. The
strength of the interaction associated with the third dressing
field is reduced by another factor of 2, such that the resultant
interaction strength is d 3 5 x 3 /22 5 x 3 /4. Remember that
when the entire ladder of states is considered ~rather than just
the subgroup shown in Fig. 4!, all doubly dressed states are
split by this amount. There are eight triply dressed states and
each give a transition of equal strength to the third atomic
level. Hence, the Autler-Townes spectrum should be comprised of eight lines of near equal intensity, positioned at
1
2 (6 d 1 6 d 2 6 d 3 ).
A fourth field, resonant between the triply dressed states,
can be applied in a number of ways, but for simplicity only
one is illustrated in this paper ~Fig. 5!. It is readily anticipated that each spectral feature in Fig. 4 is split by the ap-
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FIG. 4. Field configuration and spectra for four separate cases
involving three strong pump fields, with x 1 5120.6 kHz, D 1 50
kHz, x 2 560.0 kHz, D 2 52121.0 kHz. The spectra ~i!–~iv! correspond to the driving fields resonant with transitions ~i!–~iv! shown
in the upper figure. A minimum number of dressed states and driving fields are shown. Also for clarity only one of the eight transitions to the third level u c & is indicated.
plication of the fourth field. The general form for the splitting induced by field i when applied resonant with the i21
dressed states is given by d i 5 x i /2i21 . This yields a sixteenline spectrum with features at 21 (6 d 1 6 d 2 6 d 3 6 d 4 ). The
validity of such expressions relies on considering the resonant interactions only and neglecting off-resonant interactions. Consequently the approach is less valid when the detunings are smaller, as occurs with a larger number of fields.
There will be a decreasing denominator associated with the
off-resonant terms and their contribution in a perturbation
series expansion for the energies will become significant.
Neglecting these terms will clearly result in inaccuracies.
Our experimental configuration is realized by exciting the
DI561 magnetic dipole transitions associated with a
nuclear spin I51 system of a nitrogen atom at a nitrogenvacancy color center in diamond. The transition frequency
for the u 0 & 2 u 11 & transition, corresponding to the u a & 2 u b &
transition, is v ab 54.6 MHz, and the transition frequency for
the u 0 & 2 u 21 & transition, corresponding to the u a & 2 u c &
~probe! transition, is v ac 55.4 MHz. The effect of the probe
field was then measured using the coherent optical method of
Raman heterodyne detection @6#. The experimental details
and a discussion of the hyperfine levels of the nitrogen vacancy center can be found elsewhere @7#.
Experimental spectra are presented in Figs. 2–5 and each
trace is discussed briefly. A strong on-resonance field ~field
1! is applied to the u a & 2 u b & transition with a Rabi frequency
x 1 5120.6 kHz. The applied field induces an Autler-Townes
splitting of 120.6 kHz as shown in Fig. 2. Field 2 is intro-
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FIG. 5. Field configuration and spectra for four strong pump
fields, with x 1 5120.6 kHz, D 1 50 kHz, x 2 560.0 kHz, D 2 5
2121.0 kHz, x 3 530.5 kHz, D 3 52148.5 kHz. In the dressed state
energy level diagram a minimum number of dressed states, driving
fields, and probe transitions are shown.
duced with a Rabi frequency x 2 560.0 kHz and is detuned
from resonance by the Rabi frequency of field 1. In case ~i!,
D 2 52 d 1 52 x 1 52120.6 kHz. Field 2 is then near resonant with the singly dressed transitions formed by field 1
giving the situation shown in Fig. 3. Exactly at resonance
there is a minimum splitting between the levels and, correspondingly, a minimum in the observed doublet splitting.
The resonance situation is achieved by adjusting the detuning to D 2 52121.0 kHz to minimize the spectral splitting.
The small displacement in the resonance position is due to
higher order effects, not covered by our perturbative approach, and is due to generalized Bloch-Siegert shifts @8#
arising from the bichromatic driving field. These shifts
should not be confused with regular Bloch-Siegert shifts @9#
where a second field arises from the counter-rotating term of
a monochromatic field. The Autler-Townes spectra in the
present bichromatic resonance situation is shown in the upper experimental trace of Fig. 3. It comprised of a pair of
doublets with a mean separation of 120.6 kHz determined by
the Rabi frequency of the first field and a doublet splitting of
27.2 kHz determined by the strength of the second field. This
latter splitting is close to x 2 /2560.0/2530.0 kHz, the value
predicted from our simple analysis. The four lines in the
spectrum, although clearly not equal, all have similar
strengths consistent with the earlier analysis.
When the second field is positively detuned to be resonant
with the other dressed state transition ~ii! given by a detuning
of D 2 5120.0 kHz, a similar spectrum is obtained as shown
in the lower experimental trace of Fig. 3. The splittings are
the same as for the positive detuning case and there are only
minor changes in intensity of the features. These observa-
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tions are consistent with the dressed state analysis @4#.
Four separate examples of the effect of introducing a third
field are shown in Fig. 4. To attain resonance, the detuning
of the third field is given in first order as D 3 56 d 1 6 d 2 and
values close to these detunings were found to give the minimum splittings. For example, shifts of 61.3 kHz were attained for the outer transitions ~i! and ~iv! in Fig. 4 and 60.7
kHz for the inner transitions ~ii! and ~iii!. More significantly
it can be seen that the spectrum is very close to that predicted
by the approximate analysis, with each spectrum exhibiting
four sets of doublets with almost equal splitting. The Rabi
frequency of the third field was x 3 530.5 kHz and the observed splitting was d 3 57.5 kHz, very close to the predicted
value of d 3 5 x 3 /457.9 kHz. The eight lines are again not of
equal intensity but are all readily detectable.
One example of a resonance with four fields is illustrated
in Fig. 5. This is shown for the case where each field has a
minimum frequency necessary to attain a resonance between
dressed states. The detuning of the fourth field was D 4 5
2155.1 kHz, 0.4 kHz from that predicted from the splittings
of the three field Autler-Townes spectrum. Consistent with
expectations, the three field features are further split into
doublets by the application of the fourth field and the splitting of 1.7 kHz is close to the 15/851.9 kHz expected for
field strength of x 4 515 kHz. Most of the sixteen features are
observed in the expanded scales in Fig. 5. Several of the
features occur as shoulders on the side of other lines and
clearly the features are far from equal intensity. This indicates that the approximations used in the simple analysis are
now questionable. The simple perturbation approach gives
an unreliable estimate of the eigenstates but still gives a reasonable estimate of the eigenvalues.
The breakdown of the perturbation approach occurs when
the effects of the fields on the off-resonant transitions become significant. Should it be possible to increase the separation of the dressed states, the perturbative expressions
would be valid for an even higher number of fields. Increased splitting can be achieved through the use of stronger
fields, however all fields would have to be increased proportionally, necessitating a very large value of the first field. It is
not inconceivable that this idea could be tested. However, in
the present experimental system the frequency difference of
the pumping and probing transition is within 800 kHz, and
should the Rabi frequency associated with the first field
strength approach this value it would then be necessary to
consider the field interacting with a three-level system rather
than a two-level system. This would nullify the original intention of this work so far as it relates to simple extension of
the dressed state description for a two-level system interacting with polychromatic photon fields.
In conclusion, we have observed the spectra obtained by
polychromatic excitation of a two-level atom with the fields
applied resonantly with transitions between dressed states.
This was achieved by exciting transitions between multiply
dressed states. When placed on resonance between dressed
states, each field induces a splitting in the total system, redressing the states. Our studies and analysis have demonstrated the usefulness and physical basis of the dressed states
formalism for multiple fields.
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