Abstract
The influence of the dephasing on interference is investigated theoretically and experimentally in three-level electromagnetically induced transparency systems. The nature of the interference, constructive, no interference or destructive, can be controlled by adjusting the dephasing rates. This new phenomenon is experimentally observed in meta-atoms. The physics behind the dephasing-induced control of interference nature is the competing between stimulated emission and spontaneous emission. The random phase fluctuation due to the dephasing will result in the correlation and anti-correlation between the two dressed states, which will enhance and reduce the stimulated emission, respectively.
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Introduction
Coherent processes based on the quantum interference in three-level media have been investigated for many years1,2,3,4,5,6,7,8,9,10,11 and yielded many novel and unexpected phenomena, for example, electromagnetically induced transparency (EIT)3,4,5, lasing without inversion (LWI)6,7,8 and spontaneous emission cancellation9,10. Alkali metals in gas phase were employed in the first and many recent experiments3,11, for their simple electronic level structure and long-lived coherence. Later experiments were also conducted with solid system, such as doped solid12, quantum dots13 and superconducting circuits14. The EIT brings the absorption cancellation (or reduction) at the resonant frequency of a transition and gives rise to steep dispersion, as well as greatly enhanced nonlinear susceptibility in the spectral region of induced transparency of the medium5. Now, EIT has become a crucial technique for its potential applications such as slowing of light15, optical storage16, quantum information processing17 and optical diodes18.
Two methods, bare states and dressed states3,5, are used in analyzing the EIT in closed three-level atoms, where a weak field probes a transition between two level and a strong field couples the upper level of the transition to the third level. In the bare state method, the absorption reduction is due to multiple pathways through the strong field coupled transition many times19. In the dressed state method, the EIT results from a combination of Autler-Townes splitting of the two dressed states and destructive interference in the probe absorption due to the dressed states20. Most of the studies on the upper-level coupled three-level atoms claim that there is only destructive interference, while for the lower-level coupled three-level atoms, most claim no interference5,21. Comparisons between the EIT and Autler-Townes splitting show that the quantum interference is important when the Rabi frequency of the coupling field is at the order of decay rates21,22,23. When the Rabi frequency is much larger than the decay rates, the two peaks of the absorption spectrum are well separated and the difference between the EIT and the Autler-Townes splitting (without interference) is very small, especially for the resonance absorption. However, the influence of the dephasing on the interference nature has not been discussed. In Ref. 24, the Author gave an approximate equation under the condition of the Rabi frequency of the coupling field much larger than the decay rates, from which he found constructive (destructive) interference for lower-level (upper-level) coupled three-level atom, if the dephasing is neglected. Although, he did not explicitly discussed the transfer of the interference nature (between constructive and destructive), this equation indicates that the transfer is possible by using the dephasing rates. Furthermore, we would like to reveal the physics behind the control of interference nature.
The classical analogies of the EIT based on artificial meta-atoms recently have attracted a lot of attention25,26,27,28,29,30,31,32,33 due to easy experimental demonstration and potential applications, such as slow light25,26, low-loss metamaterial27,28, optical storage30 and sensing31. For the meta-atoms, not only the coupling strength (corresponding to the strength of the driving field), but also the intrinsic parameters of the systems can be adjusted in a relative easy way, such as the dephasing rates by using variable resistors34,35. Therefore, meta-atoms provide a flexible experimental platform to simulate the quantum interference phenomena in multi-level atomic systems.
In this work, we discuss the systematical control of the quantum interference nature by changing the dephasing rates in the EIT three-level system (the upper level coupled by a strong field) with the Rabi frequency of the coupling field at the order of the decay rates. Our dephasing-induced control of interference is established with the two dressed states. The dynamic equations for the two dressed states clearly show that the correlation between the two dressed states and consequently the nature of the quantum interference, is determined by the decay rates, so that one can have constructive, destructive or no interference in the atomic absorption spectrum by simply changing the decay rates, particularly the dephasing rates. Here we also report the first experimental observation of the dephasing-induced control of the interference nature in the EIT three-level meta-atoms. The dephasing-induced transfer from constructive interference to destructive interference is demonstrated. Our results provide a clear understanding of the nature of quantum interference in three-level EIT systems and pave a new way to control all kinds of interference by adjusting the dephasing rates.
Results
Dephasing-induced control of interference nature
Consider the upper-level coupled three-level systems, which can be EIT-
[see Fig. 1(a)] or EIT-cascade [see Fig. 1(b)] schemes. The three-level atom is driven by a strong field with frequency 
and Rabi frequency Ω and is probed by a weak field with frequency 
and Rabi frequency Ωp. The Hamiltonian is 
. Setting 
and 
(resonant driving), the Hamiltonian in rotating frame can be written as 
for the 
scheme. The Hamiltonian for the cascade is the same with 
replaced by 
. The density matrix equations for 
scheme in bare states are
Three-level schemes with resonant driving.
(a) EIT-
type. (b) EIT-cascade type. 
and 
are the Rabi frequencies of the coupling and probing fields, respectively. 
denotes the frequency detuning.
where 
is the population decay rate from 
to 
and 
is the off-diagonal decay rate of 
. Note 
. The equations for the cascade scheme are the same with 
replaced by 
and the second equation replaced by 
.
The imaginary part of the atomic linear susceptibility, or the absorption, is proportional to the real part of 
, which can be obtained by perturbation method and is the same for both schemes,
Let us consider the dressed states 
and 
. The density matrix equations in the dressed states for both schemes can be obtained from Eqs. (1),
where 
. With the zeroth order solution 
and 
, we can obtain
and the absorption becomes 
.
If 
, Eqs. (3a) and (3b) are two independent equations and have no correlation between them and Eqs. (4) tell us that the emission spectrum is a sum of two independent Lorentzian peaks with the same linewidth 
and there is no interference. Y is responsible for the correlation between 
and 
and the interference in the absorption. The energy difference between the two dressed states is 
determined by the driving field. We can consider that the parameter 
, which can be written as 
(−1 < p < 1), describes the correlation between the two effective dipole moments (
and 
) of the two transitions. Such dipole correlation represented by 
will result in the quantum interference phenomenon9. For 
(
), we have the two dipole moments orthogonal and no interference. For 
(
), we have an acute angle between the two dipole moments and the absorption around 
is less than the sum of the two Lorentzians, which means the destructive quantum interference in the absorption. For 
(
), we have an obtuse angle between two dipole moments and the absorption around 
is more than the sum of the two Lorentzians, which means the constructive interference in the absorption. As 
, we have 
. We can see that 
and 
have different roles in the interference: destructive (
and 
) and constructive (
and 
), respectively.
As Y, which depends on the decay rates, is responsible for the interference in the absorption, it is clear that only the stimulated absorption from the lower level 
to 
and from 
to 
(from one state to two different states) could not form the pair of the interference pathways. The two interference pathways must include spontaneous decay process. Therefore, the two pathways, which form the interference pair, are stimulated absorption (from 
to 
and 
) and then spontaneous emission (from 
and 
to 
), see Fig. 2. It is the interference of the spontaneous quenching together with the stimulated absorption that makes the electromagnetically induced transparency.
The interference pathways in EIT system.
The two pathways, which form the interference pair, are stimulated absorption (from 
to 
and 
) and the spontaneous emission (from 
and 
to 
).
For the 
scheme we have 
and 
, with 
and 
the dephasing rates for 
and 
, respectively. If the dephasing rate 
is very small compared with 
, we have 
, leading to almost complete destructive interference at 
. If we have very large dephasing rate 
compared with 
, which can be realized by adding some collision mechanism with level 
(not level 
), we will have 
, resulting in almost complete constructive interference at 
. By changing the dephasing rate 
from small to large, we can have from almost complete destructive interference to no interference and then to constructive interference. In Fig. 3, we plot the real part of 
for the 
scheme with different 
from 
to 
with 
and 
. The blue curve is a sum of two Lorentzians for 
. From Fig. 3 we can see clearly the transition from complete destructive interference (red curve) to constructive interference (orange curve). Similar curves can be found for the cascade EIT scheme, from almost complete constructive (
) to almost complete destructive (
) interference by changing only the dephasing rates.
Calculated absorption spectra for upper-level coupled three-level system, which is proportional to the real part of 
.
The magnitude of absorption at the resonance frequency gradually decreases with 
. For 
, the absorption spectrum equals to the sum of two Lorentzian peaks with the same linewidth 
(Blue). For 
(
), constructive (destructive) interference of the absorption is observed at the resonance frequency.
We ask ourselves why 
and 
play the different roles in the interference and what is the mechanism to understand the dephasing-induced control of interference. The decay rates 
and 
are related to the population decay and dephasing of level 
and 
, respectively. The decay 
is caused by population and dephasing reservoirs of 
, while decay 
is caused by population and dephasing reservoirs of 
. The reservoirs of 
(or 
) result in random phase changes of 
(or 
). The random phase changes of 
and 
due to the reservoirs of 
are correlated by the strong field, so that the two random phase changes are of the same magnitude and the same sign because of 
(the common plus sign before 
). Due to the correlation of the random phase changes for 
and 
, the system from 
to 
and 
by stimulated absorbing a probe photon can stimulatedly emit the same probe photon with the same frequency and phase from to 
and 
to 
, which has no contribution to the absorption. It is these stimulated emission processes that prevent the spontaneous emission, so that the contribution of the reservoirs of 
to the spontaneous emission is reduced or eliminated. This is why we have destructive interference. Similar discussion can be made for the reservoirs of 
, which will result in anti-correlation (same magnitude with opposite signs) for the random phase changes, because of 
(the opposite signs before 
), which leads to reduction or inhibition of the stimulated emission and enhances the spontaneous emission, so that we have constructive interference in the absorption. For 
(Y = 0), the correlation and the anti-correlation cancel each other and we have no interference.
Experimental demonstration in meta-atoms
For an atomic system, the adjustment of the dephasing rates is limited because it is difficult to get proper parameters by direct tuning the atomic collisions36,37,38. Here we resort to the classical analog of EIT in meta-atoms to give the experimental demonstration of the dephasing-induced control of interference nature.
It has been shown that the dynamic equations for the classical EIT in meta-atoms are given as26,33,39,40
where 
is the input field with frequency detuning 
which will excite the mode 
(but not the mode 
) and 
is the near-field coupling strength between the two resonant modes in the meta-atom (
’s role is same as the Rabi frequency 
in atomic EIT system). 
is a “bright” mode that has outputs in both forward and backward directions with a scattering loss rate 
, while 
is a “dark” mode that has no output and no scattering loss. The two modes have the same resonant frequency and suffer decay with different dephasing rates 
and 
, respectively.
After some derivations, we obtained that
where 
, 
, 
is the reflection coefficient and the transmission coefficient is 
. The dependence of 
on the decay rates of the resonant modes (
and 
) is the same as that in 
on the atomic decay rates, see Eqs. (2) and (4). Therefore, the real part of 
corresponds to the absorption in the atomic EIT system. We can measure r versus 
to simulate the dependence of the absorption in the atomic EIT system on 
.
Our experiments are conducted by using meta-atom in microwave range. The experimental setup is shown in Fig. 4. The meta-atom is composed of two coupled resonators. The bright resonator (
) is a copper branch with the length of 
and it is connected with the main-strip by the resistor 
, as indicted in the middle of Fig. 4. The dark resonator (
) is composed of two metal split rings with the dimensions of 
, which are located at the two sides of the first resonator. It has been demonstrated that the response of this configuration composed of the bright and the dark resonators can be regarded as the classical analogue of the EIT26,27,28,29,34,35. Here the width of the metal copper wires is 
. The gap size of the two rings is 
. The frequencies of the two resonant modes are designed to be the same at 23.56 GHz. The dephasing rates of the two resonators, 
, can be adjusted by two resistors 
, respectively. By putting the two resonators close to each other with a distance of 
, we can introduce the near-field coupling 
between the two modes. When a microwave of frequency 
propagates along the main-strip (incident from the left), we have input to the bright resonator (
) which excites 
. The dark resonator is far away from the main-strip, so that no input for the dark resonator. The dark mode 
is excited by the coupling between the two modes. The motion of the resonant modes is described by Eq. (5). After doing the full-wave simulations of the meta-atom with the chosen geometric parameters, we have 
, 
and 
, 
39. Then we can adjust 
and 
to have different 
and 
.
Experimental setup to simulate the dephasing-induced control of interference in atomic EIT system.
Transmission and reflection coefficients are measured with the microwave network analyzer. The EIT meta-atom consists of a “bright” resonator, a “dark” resonator, the pair of split rings. The near-field coupling strength between the two resonators is determined by the separation 
and the dephasing rates of the two resonators, 
, can be adjusted by two resistors 
, respectively. Geometric parameters can be found in the Main Text.
In Fig. 5, we present the measured spectral of the real part of 
(rather than the equivalent quantity 
) with different dephasing rates given by resistors 
. This is mainly for two reasons: (i) Compared to the reflection coefficient r with respect of the mirror plane of the structure, the transmission coefficient t is relatively easy to measure. (ii) The slight asymmetry of the sample due to the fabrication leads to the difference between the reflections for the incident from left side and right side, while the transmissions are reciprocal and uniquely determined. Here we fix 
, which makes 
be a constant, 
and all of the frequencies and dephasing rates are normalized with 
. By changing R1 and R2 simultaneously, we can make 
from 
(
) to 
(
), that is to say, from a constructive interference to a destructive interference. The measured spectra of 
in Fig. 5 show clearly the phenomenon of the dephasing-induced control of interference, namely, the transition from the constructive to the destructive interference with the increasing of 
. For the case of 
(
and 
), there is no interference and the measured spectrum approaches to the sum of the two Lorenzians, see the blue curve (the theoretical calculation is also plotted for comparison, see the gray dashed curve), with the resonance centers 
and the common line-width 
. Here non-zero value of 
at 
(red curve in Fig. 5) is due to a little inevitable dissipation from the material and the roughness. Although there is a little difference between our measurements and theoretical calculation due to the fabrication tolerance, as well as the effects from the high-order modes and the imperfect match at input/output ports (as these unfavorable factors are not considered in theoretical calculation), our experiments have no doubt simulated the constructive, destructive and no interference in EIT three-level system by adjusting only the dephasing rates in meta-atom.
Measured spectral of the real part of 
with different dephasing rates given by resistors R1,2.
The real part of 
corresponds to the the real part of 
. As a comparison, the theoretical calculation for 
is given with the gray dash line. Our measurements consist with the calculations in Fig. 3 reasonably well.
Discussion
The new phenomenon, the dephasing-induced control of interference nature has been theoretical investigated and experimentally observed in the three-level EIT system. We find that the nature of the interference is dependent on the decay rates, but not related to the strength of the driving field. The random phase fluctuation due to the dephasing will result in the correlation and anti-correlation between the two dressed states, which will enhance and reduce the stimulated emission, respectively. The physics behind the dephasing-induced control of interference is the competing between stimulated emission and spontaneous emission.
Methods
The samples are all fabricated on copper-clad 0.787-mm thick Rogers RT5880 substrates using laser direct structuring technology (LPKF ProtoLaser 200). Transmission and reflection properties were obtained directly from the microwave vector network analyzer (Agilent N5222A). In addition, a commercial software package (CST Microwave Studio) is used in designing the samples.
Additional Information
How to cite this article: Sun, Y. et al. Dephasing-Induced Control of Interference Nature in Three-Level Electromagnetically Induced Tansparency Systems. Sci. Rep. 5, 16370; doi: 10.1038/srep16370 (2015).
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Acknowledgements
This work was supported by the National Key Basic Research Program (973) of China (Nos. 2012CB921601, 2013CB632701 and 2011CB922001), NSAF (Grant No. u1330203) and National Natural Science Foundation of China (Grants Nos.U1330203, 11174026, 11474221, 11234010 and 11204217). YS and HC also acknowledge the support from the Fundamental Research Funds for the Central Universities.
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Y.S. and Y.Y. designed and carried out the experiment, while S.Z. and H.C. with the help of Y.S. worked out the theory. All authors contributed to the writing of the manuscript.
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Sun, Y., Yang, Y., Chen, H. et al. Dephasing-Induced Control of Interference Nature in Three-Level Electromagnetically Induced Tansparency Systems. Sci Rep 5, 16370 (2015). https://doi.org/10.1038/srep16370
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DOI: https://doi.org/10.1038/srep16370
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