Circuit Analog of the 4-Level Atomic System

In the literature, new schemes based on optical and photonic scenarios, such as planar metamaterial analogue of EIT for plasmonic sensing, plasmonic analog of EIT , circuit analog of EIT as well as EIT-like effect in micromechanical resonators, have been suggested or experimentally demonstrated. Although EIT is a quantum optical effect arising in atomic systems, yet it has various analogs in classical physics realm. Such analogs can be referred to as “classical EIT” effects or “EIT-like” phenomena, which can also exhibit “destructive interference” between two (or among three) resonant systems. It should be noted that almost all these classical analogs of EIT are those of three-level systems.As a four-level EIT system can exhibit a more significant dispersion sensitive to the probe frequency than a conventional three-level system, the quantum optical properties of four-level EIT (such as double-control quantum interference, transient turn-on and -off dynamics) as well as its application to photonic device design (e.g., photonic switches and logic gates), have focused intensive attentions of many researchers. In this paper we shall develop a scenario for investigating the circuit analog of quantum coherent effects, including two-level resonance, three- and four-level electromagnetically induced transparency.

Since we will focus on the stimulation of EIT in a medium composed of a four-level tripod-configuration atomic system interacting with three laser fields (see Fig. 2.6 for the schematic diagram), here we shall address the intriguing optical behavior of this four-level EIT atomic vapor (the two- and three-level resonant systems will be the special cases of such a four-level EIT atomic system). Considera four-level atomic system with three ground levels |1>, |2>, |2’> and one excited level |3>. This atomic system interacts with the electric fields of the probe waves and two control light waves, which drive the |1>-|3>, |2>-|3> and

|2’>-|3> transitions, respectively. As there are two control fields and one probe field interacting with the four-level system, the multilevel transitions, where the constructive and destructive interference among various transition pathways will arise, would give rise to new novel optical behavior of the EIT medium. If, for example, the ratio of the intensities (characterized by the squares of the Rabi frequencies, i.e., and ) of the two control fields are taken to be certain proper values, the double-control destructive interference between |2>-|3> and |2’>-|3> transitions would occur. Then, it seems that the two levels |2> and |2’> as well as the two control fields ( , ) are absent, and hence the

 *_{c c}  ^*_{c' c'}

_c _c'

87

level system would be reduced to a simple two-level system {|1>, |3>}. Thus, the four-level EIT vapor is no longer transparent to the probe field.

The Schrödinger equation that governs the density matrix elements of thefour-level atomic system is given by

(5-1)

(5-2)

(5-3)

Here, and stand for the spontaneous emission decay rate and the collisional dephasing rates (nonradiative decay rates), respectively. and denote the Rabi frequencies of the two control fields and the probe field. They are defined by

and respectively, with and the

slowly-varying amplitudes (envelopes) of the control fields and the probe field. The frequency detunings of the three applied optical fields are defined as , , with and , the mode frequencies of the probe and control fields, respectively. In general, the population ( ) in the ground level is almost unity (i.e., ). It should be noted that the three-level EIT system is a special case of the four-level system. If, for example, one of the control fields, say, , is switched off (i.e., ), then the present four-level atomic system will be reduced to a three-level system.

The circuit analog to the atomic system of Fig. 2.1 is shown in Fig. 5.2(a), where the circuit mesh consisting of the inductor L and the capacitors C , C and _x C is used to _y simulate the two-level probetransition (i.e., |1>-|3> transition driven by the probe field), and the resistor R leads to the oscillator losses that represent the atomic spontaneous emission decay (from the upper level |3> to the lower levels) in the EIT atomic system. The two level pairs and in Fig.2.6 are modeled by the two resonant loops on the two sides: specifically,the resonant loop formed by the inductor L and the capacitors ₁ C , ₁ C_x

 

^*c

21

2 p c 21 31

i

d i ,

dt 2

  ^     ^  ^ 

 

^*c'

2'1

2' p c' 2'1 31

i

d i ,

dt 2

  ^     ^  ^ 

 

^p

31 c c'

3 p 31 21 2'1 11

d i i i

i .

dt 2 2 2

        ^   ^   ^ 

3  ₂, _2'

c, c'

  _p

c 32Ec/ ,

   _{c' 32'}E_c'/  _{p 31}E_p/ , E_c, E_c' E_p

31 

 _p  _p

32 

 _c  _c  _c' _32'_c' _p _c _c'

11

1 ₁₁1

c' c' 0

 



^{2 , 3}

 

^{2 ' , 3}



88

on the left-handed side and the resonant loop formed by the inductor L and the capacitors ₂ C , 2 C on the right-handed side are the analogs of _y and , respectively.The two resistors R₁ and R in the circuit represents the population decay ₂ from the upper levels to the ground level (due to collisional dephasing effect).

The capacitors C and _x C , which belong to their respective circuit meshes, can model the _y contribution of the two control fields, i.e., 1 /C and 1 /_x C would give rise to the effect _y similar to what thetwo Rabi frequencies _c and _c have contributed. In this sense, the frequency-tunable voltage source V in the circuit stands for the probe field in the atomic _p EIT.

Since a two-level resonant atomic system can be simulated by a simple circuit, three- and four-level electromagnetically induced transparency (EIT) that occur due to light-atom interaction can find its classical counterpart in circuit analog. As the optical response of an EIT atomic medium (including atomic vapors and semiconductor-quantum-dot dielectrics) can be controlled via tunable quantum interference induced by applied external control fields, in the scheme of circuit analog, such a controllable manipulation is achieved via capacitor coupling, where two loops are coupled by a capacitor that can represent the applied control fields in atomic EIT. Both numerical simulation and experimental demonstration of three- and four-level EIT were performed based on such a scenario of circuit analog. The classical “coherence” relevant to quantum interference among transitions pathways driven by both probe and control fields in EIT atomic systems has been manifested in the present circuit analog of EIT.

The quantum coherent control via atomic phase coherence has led to a number of physically interesting phenomena such as electromagnetically induced transparency (EIT) and the effects that are relevant to EIT, including inversionless light amplification, cancellation of spontaneous emission, multi-photon population trapping, phase coherence control as well as EIT-induced negatively refracting materials. EIT is such a quantum optical phenomenon that if one resonant laser beam propagates in a medium (e.g., an atomic vapor or a semiconductor-quantum-dot material), the beam will get absorbed; but if two resonant laser beams instead propagate inside the same medium, neither would be absorbed. Thus the opaque medium becomes a transparent one. Such an interesting optical behavior would lead to many applications, e.g., designs of new photonic and quantum optical devices. Since it



^{2 , 3}

 

^{2 ' , 3}



2 , 2 ' 1

LC

LC

89

can exhibit many intriguing optical properties, EIT has attracted extensive attentions of researchers in a variety of areas of atomic physics, optics, and condensed state physics. For example, some unusual physical effects associated with EIT include ultraslow light, superluminal propagation, and optical storage with atomic vapors, some of which are expected to be beneficial (and powerful) for developing new technologies in quantum optics and photonics.

Although EIT is a quantum optical effect arising in atomic systems, yet it has various analogs in classical physics regime. Such analogs can be referred to as “classical EIT” effects or “EIT-like” phenomena, which can also exhibit “destructive interference” between two (or among three) resonant systems. In the literature, new schemes based on optical and photonic scenarios, such as planar metamaterial analogue of EIT for plasmonic sensing, plasmonic analog of EIT, circuit analog of EIT as well as EIT-like effect in micromechanical resonators, have been suggested or experimentally demonstrated. It should be noted that almost all these classical analogs of EIT are those of three-level systems. As a four-level EIT system can exhibit a more significant dispersion sensitive to probe frequency than that in a conventional three-level system, the quantum optical properties of four-level EIT (such as double-control quantum interference, transient turn-on and -off dynamics) as well as its application to photonic device design (e.g., photonic switches and logic gates), have captured intensive attentions of many researchers. Obviously, the impact would be enormous if we can experimentally verify the circuit analog of both three- and four-level EIT. In this paper, we shall report our work on such a classical EIT-like effect, including their classical analog of relevant quantum coherence involved, e.g., destructive quantum interference among transitions pathways driven by both probe and control fields.

We shall now concentrate our attention on the influence of the external control fields on the three- and four-level atomic population, which is analyzed based on the concepts of quantum interference and dark states. The three-level EIT occurs when the condition,

, of two-photon resonance is fulfilled. One can see from the first equation in (1) that the upper level would be empty, i.e., the steady solution (Note that the collisional dephasing rates depends on the pressure of the vapor, and it would be negligibly small in a dilute atomic vapor). Thus, the two lower levels , of the three-level system would form a dark state: specifically, it follows from the third equation in (1) that the control and probe fields (characterized by the Rabi frequencies and ) agree

  _{p c}

3 ₃₁0

2

1 2

_c _p

90

with the condition of the destructive quantum interference, . This, therefore, means that the probability amplitudes of levels |1 and | 2 are N_c and

N p

  , respectively, and the dark state of the atomic system is given by



^{c p}



| Dark =N  |1   | 2 with the normalized coefficient . The minus sign in the dark state | Dark represents an effect of destructive quantum interference between the two transition pathways |1  | 3 and | 2  | 3 . This leads to an effect of electromagnetically induced transparency to the probe field, since there is no effective interaction between the dark state and the applied fields (control and probe fields)^[1], and hence no net transition from the ground level |1 to the upper level | 3 occurs (i.e.,

). In the four-level system, the concept of dark state would be generalized, i.e., a three-level dark state that consists of the three lower levels ( |1, | 2 and | 2) would be generated under certain proper conditions. In this case, the double-control EIT would yield.

In addition to this three-level dark state, there is a new state (characterized by

21/ 2 1 c / c

  _   _ ) that leads to a double-control destructive quantum interference between |1 | 2  and |1 | 2 transitions, i.e., levels | 2 and | 2 form a new dark state, i.e., which can be expressed by ^N



^{c 2  c 2}



with the normalized coefficient

* *

c c c c

1 /

N      _{ } . Under this condition, it seems that the two control fields

as well as the two levels and are not present, and hence the four-level system



|1 , | 2 , | 2 , | 3   



is reduced to a two-level system



^{|1 , | 3 }



. Since now the steady density matrix element of the |1  | 3 transition driven by the probe field is

 

31 i p 11/ 2 3 i p

    ^    ^ (an expression for a typical response of two-level resonant absorption), an effect of significant absorption to the probe field will arise because of the double-control destructive quantum interference between the two transition pathways

|1  | 2 and |1  | 2 . In a word, the multilevel atomic coherence (various quantum interference effects) can be controllably manipulated by adjusting the intensities (somewhat equivalent to the Rabi frequencies) of the control fields.

11 0

 

_{c 21} _p 

* *

1/

N      _{c c p p}

₃₁0

 _c, _c'

2 2 '

91

Fig. 5.1 The circuit analog of the three- and four-level quantum coherent effects. The circuit loop containing L C corresponds to the _p, _p |1 | 3 transition pathway in the atomic system. The two circuit loops on the left- and right-handed sides are analogs of the

| 2 | 3 and | 2 | 3 atomic transition pathways, respectively.

We are now in a position to utilize a circuit to simulate the above quantum interference effects. In the circuit analog of EIT (see Fig. 5.1 for the schematic diagram), if the levers of Switches X and Y join up simultaneously with contact 1 and with contact 3, respectively, a simple two-level resonant system is made via such a circuit analog (in fact, the two-level atomic system can in principle be simulated by any classical oscillating systems). If contacts 2, 3 in both Switches X and Y, or contacts 1, 2 in both Switches X and Y, are connected, an analog of a three-level resonant system yields, and if contacts 2, 3 of Switch X and contacts 1, 2 of Switch Y are simultaneously connected, this will make a three-loop circuit for analog of a four-level resonant system. The dynamical behavior of the three-loop circuit (shown in Fig. 2.6) is governed by

2

1 1 1

1 p

2

1 1 1 1

1 1 1 1

x x

d i R di

i i

dt L dt L C C L C

 

      

  (5-4)

2

2 2 2

2 p

2

2 2 2 2

1 1 1 1

y y

d i R di

i i

dt L dt L C C L C

 

       (5-5)

2

p p p p

p 1 2

2

p p p p p p

1 1 1 1 1 1 1

x y x y

d i R di dV

i i i

dt L dt L C C C L C L C L dt

 

          (5-6)

It can be clearly seen that the mathematical structure of the circuit equation (2) is analogous to that of the density matrix equation (1) of the four-level atomic system:

92

specifically, we can have the relations i_p ₃₁, i₁₂₁, i₂ _{2 1} and 1 /C_x _c, 1 /C_y  c_. The term ofdV_p/dt , i.e., the voltage signal source, corresponds to the Rabi frequency, , of the probe field.

The typical parameters in the circuit are chosen as follows:C₁2.2 F , L₁ 500 H ,

2 2.2 F

C   , L₂ 500 H ,C_p 100 F , L_p 22 H , and R_p 10. The amplitude of the voltage signal source is V_p(max) 0.1 V. The resistances R and ₁ R in numerical ₂ simulation and experimental realization are quite small (this corresponds to the fact that the collisional dephasing rates ₂, _2 in a dilute atomic vapor are negligibly small). The behaviors of the forward transmission coefficient of the circuits (including the circuit analog of two-, three- and four- level resonance) are plotted in Fig. 5.2(a) to Fig. 5.2(f) by making use of ADS (Advanced Design System, Agilent software). Here various capacitances

representing the applied control fields in atomic ensembles are chosen for the ADS simulation (1 /C and 1 /_x C correspond to _y and , respectively). From Fig. 5.2(a), 5.2(b) one can find that for the curve of the parameter 20 log S ₂₁ in the circuit analog of the two-level system (i.e., the simple R L C circuit), there is only one cuspidate point (a _{p p p} downward peak) indicating two-level resonant absorption, while in the circuit analog of three- and four-level systems, there are two and three downward cuspidate points, respectively. These downward peaks can be interpreted as “Autler-Townes effects”. It follows that the circuit composed of two resonators (L C ,_{p p} L C ) via capacitor coupling has _{1 1} an Autler-Townes doublet, and the circuit composed of three resonators (L C ,_{p p} L C ,_{1 1} L C ) _{2 2} has an Autler-Townes triplet. Thus, there is only one EIT transparency window between the two downward cuspidate points in Autler-Townes doublet (corresponding to the circuit analog of the three-level system) and two EIT transparency windows among the three downward cuspidate points in Autler-Townes triplet (corresponding to the circuit analog of the four-level system). It should be noticed that for the curves of the parameter 20 log S ₂₁ in Fig. 5.2(c), 5.2(d), 5.2(e), 5.2(f), the Autler-Townes triplet in the four-level system will finally become an Autler-Townes doublet since the value C approaches _x C . _y

As an illustrative example, in what follows, let us point out briefly the mechanism that leads to the presence of the transparency windows in the circuit analog of three-level EIT. If,

_p

x, C

Cy

_c _c'

93

for example, contacts 2 and 3 in both Switch X and Switch Y are connected simultaneously, the two resonant systems (L C ,_{p p} L C ) are coupled via the capacitor _{1 1} C . Such a capacitor _x coupling would dramatically modify the resonant behavior of L C system: specifically, _{p p} the electric current passing through R that should have increased to a quite large value at _p the resonant frequency will now be small (i.e., suppressed) because of the destructive interference between the two resonant systems (L C and _{p p} L C ). _{1 1}

(a)

(b)

(c)

(d)

94 (e)

(f)

Fig.5.2 The ADS simulation results of Autler-Townes doublet (triplet) and EIT transparency windows for the circuit analog of two-, three- and four-level quantum coherent effects (corresponding to the various capacitances chosen for

C ). There are one, two and three downward cuspidate points in the curves of x

for the analog of two-, three-, and four-level systems, respectively.

In (d) the Townes triplet in the four-level system is reduced to the Autler-Townes doublet because the two coupling capacitors have the same capacitances (C_x C_y).

5.2 Experimental realization of three- and four-level quantum coherence with circuit (GPIB Automatic Test System)

Since the electric current through the resistor R is analogous to the density matrix _p element ₃₁ due to the |1 | 3 transition driven by the probe field, the behavior of the voltage drop across the resistor R would also manifest its quantum counterpart in EIT via _p the circuit analog.

20 log |S21|

95

Fig. 5.3 The configuration of experimental setup for measuring the voltage .

In order to achieve the analog of the EIT systems, we designed a configuration of experimental setup shown in Fig. 5.3. Let the function generator sweeps its frequencies from 20 kHz to 1 MHz (i.e., X external synchronized input to the oscilloscope), and then the oscilloscope functional stepping button was rotated into its “Diff.” mode to measure the voltage difference between Y and ₁ Y , which are the probed voltages of node 1 and node ₂ 2 of two terminals of (see the circuit in Fig. 5.1) . Here, the oscilloscope was used as an XY recorder, where its primary time base (transverse axis) has been taken over as a frequency base. To obtain the analog of the three-level EIT, one of the two Switches (X and Y in the circuit shown in Fig. 5.1) were pressed to make the two contacts 7 and 8 in Switch X and make the two contacts 0 and 4 in Switch Y touched or to make the two contacts 1 and 8 in Switch X and make the two contacts 0 and 11 in Switch Y touched, respectively. The experimental result, i.e., the measured voltage V of the resistor ₁₂ R , is presented in Fig. _p 5.1, where the tunable capacitance C representing the contribution of the control field _x (with Rabi frequency _c) is chosen as C_x 220 pF, 1 nF, 10 nF, and 20 nF, respectively. It should be noted that the resonant frequency of the signal source, where the voltage V ₁₂ takes a minimum value, decreases when the capacitance C increases. This is because the _x capacitance C can modify the resonant frequency _x f₀^{2 L} of the system L C . This can be _{p p} interpreted by the third equation in (2), where the resonant frequency of the systems L C _{p p} reads

2 L 0

p p

1 1 1 1 1

2 _{x y}

f  L C C C

 

     (5-7)

In Fig. 5.2(a) the resonant frequency , where the voltage (and hence the current) take the V12

R_p

f

f

96

minimal value, is 193 kHz, and the corresponding voltage is 1.7 mV. In Fig. 5(b) the minimal voltage 1.9 mV at the resonant frequency 200 kHz, in Fig. 5(c) 1.7 mV at 76 kHz, and in Fig. 5(d) 1.8 mV at 53 kHz. Note that in the three-level EIT system (Fig. 1(a)), there is only one EIT window in frequency band (see Fig. 5).

In the four-level EIT system (Fig. 1(b)), however, two EIT windows in frequency band will result from the multilevel transition pathways, where either the |1 | 2 transition driven by the control field _c or the |1| 2 transition driven by the control field would destructively interfere with the |1 | 3 transition driven by the probe field at their respective resonant frequencies. In the circuit analog of this four-level EIT system, we have used two capacitors C , _x C to represent the activity of the two control fields _y _c, _c, respectively. It follows from Fig. 5.2 that the capacitances C , _x C can make corrections _y to the resonant frequencies, where the voltage V (voltage drop) takes the minimum values. ₁₂ We should notice that there are always two minimums in the curve of V in Fig. ₁₂ 5.2(a)(b)(c). In Fig. 5.2(f), however, there is only one minimum because of the degenerate resonant frequency, namely, the capacitances of the two capacitors that couple the resonant systemsL C_{p p},L C_{1 1} and L C_{p p},L C_{2 2}respectively, are equal (i.e., 20nF, 20nF). In Fig. 5.2(a) the voltage is 2.0mV at the resonant frequency =50 kHz, and 1.7 mV at the other resonant frequency 336 kHz. In Fig. 5.2(b) 2.4 mV at 50 kHz, and 1.8 mV at 183 kHz, in Fig. 6(c) 1.7 mV at 48 kHz, and 1.8 mV at 70 kHz, and in Fig. 5.2(d) 1.7 mV at 41 kHz (only one degenerate resonant frequency). The experimental results indicate the EIT-like interference between various circuit loops (resonant systems or resonators) via capacitor couplings.

A two-level resonant atomic system can be modeled by a simple circuit, and therefore a three-level EIT system can be considered to be an ensemble composed of two coupled two-level systems. We can thus simulate the unusual resonant behavior of the three-level EIT system with a two-loop circuit via a capacitor coupling. In the same fashion, a four-level EIT system can also have a circuit analog based on a three-loop circuit via two capacitor couplings. Both three- and four-level EIT-like resonance in circuits have been numerically simulated and experimentally demonstrated. The EIT-like windows in frequency band have been verified in both three- and four-level circuit analogs. The

V12

V12 f  V₁₂

f  V₁₂ f 

_c'

_p

Cx  C_y 

V12 f

V12  f  V₁₂

f  V₁₂ f  V₁₂ f 

V12 f 

LC

LC

LC

97

dependence of the resonant frequency (corresponding to the condition of two-photon resonance in EIT) on the capacitances (for the purpose of coupling two circuits) has also been shown in the circuit analog of both three- and four-level systems. It should be pointed out that the present work is simply a low-frequency analog of EIT, and the frequency-dependence of capacitance that is out of control has unavoidably influenced our experimental results. Apparently, this scenario can be generalized to transmission analog, where such a frequency-dependence of relevant components would be avoided. We expect that such a classical EIT effect in the EIT-Like RLC circuit would find new applications in optoelectronic or photonic device designs.

5.3 Calculations forEIT-LikeRLC Circuit:

The total current i t passing through the EIT-Like circuit (see Fig. 5.1) is equivalent p

 

to the time-variant 31

 

t . We let L_p 33.4 H , C_e1C_xC_pC_y 100 F ,

p 51.7

R   denote the circuit para-meters of a two-level atomic sytem. The only one resonant frequency of the circuit can be calculated as ₀

p p

1 2.7539 Hz

2

f k

 L C

 

(2) ，

(3) ，

(4) ，

(5)

Formula substituting:

(1) ^p

 

^1Re

 

^{p p*}

P   2 V I ，

where V and _p I are the phasor forms of _p v t andp

 

i tp

 

respectively，

(2) Grounding is very important to depict on each SPICE-based software, when we use it to simulate the EIT-Like circuit. We should notice that our target is not V_R^p VS  VSV_R^p  V_Lp~_Cp_  ip



Lp  ^1 /

 

Cp ^



,the voltage drop on

equivalent bulk element which is the probing-inductor-voltage subtracting the probing-capacitor-voltage, but is the drop voltage on the probing-resistor,

LC

x 220 pF, 1 nF, 10 nF, 20 nF C 

y 220 pF, 1 nF, 10 nF, 20 nF C 

1 500 uH, 1 2.2 uF, 1 0.1

L  C  R  

2 500 uH, 2 2.2 uF, 2 0.1

L  C  R  

98

and it will be expressed as or to denote the transmission valley (transparent point).

(3) The square root of the absorption

 

A represents the dissipative coefficient，but 1

A T  R ， henceforth, a ^{min A}

 

denote the resonantvalleymeans the sametransparent peak^{max T .}

 

Fig.5.4a The transmission coefficientt can be inspected by dividing the output ₂₁ voltage to the input voltage directly( Spectrum analyzer is used)

 Rp p p

V  i R t₂₁ s₂₁

99

Fig.5.4b The transmission coefficients can be calculated by dividing the output ₂₁ wave b₂V₂/ 50to the input wavea₁V₁/ 50under a characteristic impedance ( Network analyzer is used)

Two kinds of transmission coefficientss₂₁ and t₂₁ can be acquired with or without characteristic impedance is taken into account respectively (see Appendix, page 124). A ratio between the maximum value of s₂₁ and the maximum value of t₂₁ is equal to^{2 / 2n}



^n1



^,

where, nR_p/Z₀R_p/ 50 . For example, when R_p 51.7,Z₀ 50 then

51.7 / 50

n 1.034. When the resonance undergo, the reactive part of the impedance X_p equal to zero, thens21 max_{ } / t21 max_{ } 



2n Z 0

 

/ 2n 1



Z0 j n



1



Xp0.764. EIT-like RLC circuit can be equivalent to a simple series resister R_p (which is corresponding to the spontaneous emission ₃) and inductivesusceptance X_p (which is corresponding to uncompensated frequency ₃₁)

Fig. 5.5a EIT can be represented byRp jXp

 

 and calculate outt without 50 Ohm.₂₁

Fig. 5.5b EIT can be represented byRp jXp

 

 and calculate outs with 50 Ohm. ₂₁ In which R_p 51.7 and ^jXp

 

^{  Xp}



^{,L C L Cp, p, x, y}



for 2-level EIT,

100

   

p p , p, p, x, y, 1, 2

jX   X  L C L C L C for 3-level EIT,

   

p p , p, p, x, y, 1, 2

jX   X  L C L C L C for 4-level EIT

101

5.4 Simulation using ADS and Multisim Package

Fig. 5.6 EIT-like RLC series circuit with switch S , S both OFF and has a resonant _{1 2} frequency ₀

p e

1 1.857 MHz

f 2

 L C

  where, _e ^{p x y}

p x p y x y

C C C C C C C C C C

 

而 .

Fig. 5.6.a One resonant circuit corresponding to a 2-level EIT atomic system and t₂₁ acquired. (Using MultiSim)

21

 

21 max 1

t  t  s21

 

s21 max 0.674

102 Fig. 5.6a.1 AC analysis by Multisim

2012(Magnitude versus frequency)

Fig. 5.6a.2 Network Analyzer analysis by Multisim (dB versus frequency)

Fig. 5.6.b One resonant circuit with 50 taken into account corresponding to a 2-level EIT atomic system and s acquired. (Using MultiSim) ₂₁

Fig. 5.6b.1 AC analysis by Multisim 2012(Magnitude versus frequency)

Fig. 5.6b.2 Network Analyzer analysis by Multisim (dB versus frequency)

Fig. 5.6.c One resonant circuit with 50 taken into account corresponding to a 2-level EIT atomic system and s acquired. (Using Advanced Design System)₂₁

103 Fig. 5.6c.1 s parameter sweepanalysis ₂₁

by ADS (M. Versus frequency)

Fig. 5.6c.2s parameter sweepanalysis ₂₁ by ADS (dB versus frequency)

Fig. 5.7 EIT-like RLC series circuit with switch S ON and ₁ S OFF corresponding to ₂ a 3-level atomic system have two resonant frequencies

01

p e1

1 1.92MHz

f 2

 L C

  and

 

02

p 1 e2

1 48.81kHz

2 f

L L C

  

 where,

p x y

e1

p x p y x y

C C C C C C C C C C

  , _e2 ^{p 1 y}

p 1 p y 1 y

C C C C C C C C C C

  with one transparent frequency

x 1

tr et

x 1

1 et

1 479 kHz, where, 2

f C C C

C C

 L C

  

 .

104

Fig. 5.7.a.1 Two resonant circuit without embeded50corresponding to a 3-level EIT atomic system and f₀₁, f₀₂, f acquired. (proved by MultiSim) _tr

Fig. 5.7a.1 AC analysis by Multisim 2012(Magnitude versus frequency)

Fig. 5.7a.2 Network Analyzer analysis by Multisim (dB versus frequency)

Fig. 5.7.b Two serial resonant circuitswith 50 taken into account corresponding to a 3-level EIT atomic system f₀₁, f₀₂, f acquired with lower amplitude. ( by _tr MultiSim)

105 Fig. 5.7.b.1 AC analysis by Multisim

2012(Magnitude versus frequency)

Fig. 5.7.b.2 Network Analyzer analysis by Multisim (dB versus frequency)

Fig. 5.7.c Two serial resonant circuit with 50 taken into account corresponding to a 3-levelEIT atomic system and s acquired. (Using Advanced Design System) ₂₁

Fig. 5.7c.1 s parameter sweepanalysis ₂₁ by ADS (M. Versus frequency)

Fig. 5.7c.2s parameter sweepanalysis ₂₁ by ADS (dB versus frequency)

106

Fig. 5.8 EIT-like RLC series circuit with switches S , S both ON corresponding to a 4- _{1 2} level EIT atomic system and f₀₁, f₀₂, f acquired. (proved by MultiSim) have _tr three resonant frequencies

01

p e1

1 1.92MHz

f 2

 L C

  ,

 

02

p 1 e2

1 69.82kHz

2 f

L L C

  

 and

 

03

p 1 2 e2

1 4.808kHz

2 f

L L L C

  

 

where, _e1 ^{p x y}

p x p y x y

C C C C  C C C C C C

  , _e2 ^{p 1 y}

p 1 p y 1 y

C C C C C C C C C C

  and

p 1 2 e2

p 1 p 2 1 2

C C C C C C C C C C

  with two transparent frequencies

p 1

tr_1 et

p 1

1 et

1 479 kHz, where, 2

f C C C

C C

 L C

  

 and

y 2

tr_2 et_2

y 2

2 et_2

1 50.58 kHz, where, 2

f C C C

C C

 L C

  



107

Fig. 5.8.a Three serial resonant circuit without embeded50corresponding to a 4-level EIT atomic8 system and f₀₁, f₀₂, f₀₃,f_{tr_1}, f_{tr_2} acquired. (proved by MultiSim)

Fig. 5.8a.1 AC analysis by Multisim 2012(Magnitude versus frequency)

Fig. 5.8a.2 Network Analyzer analysis by Multisim (dB versus frequency)

Fig. 5.8.b Three serial resonant circuit with embeded 50corresponding to a 4-level EIT atomic system and f₀₁, f₀₂, f₀₃, f_{tr_1}, f_{tr_2}acquired. (proved by MultiSim)

108 Fig. 5.8.b.1 AC analysis by Multisim

2012(Magnitude versus frequency)

Fig. 5.8.b.2 Network Analyzer analysis by Multisim (dB versus frequency)

Fig. 5.8.c Three serial resonant circuit with 50 taken into account corresponding to a 4-levelEIT atomic system and f₀₁, f₀₂, f₀₃,f_{tr_1}, f_{tr_2}acquired. (implemented by ADS)

Fig. 5.8.c.1 s parameter sweepanalysis ₂₁ by ADS (M. Versus frequency)

Fig. 5.8.c.2s parameter sweepanalysis ₂₁ by ADS (dB versus frequency)

109

5.5 Power Spectrum in the EIT-Like RLC Circuit

Fig. 5.9 The circuit analog of the three- and four-level quantum coherent effects.The circuit loop containing R L C, , corresponds to the |1 | 3 transition pathway in the atomic system. The two circuit loops on the left- and right-

handed sides are analogs of the | 2 | 3 and | 2 | 3 atomic transition pathways, respectively.

If the two loops on the two sides are on open-circuit state, the circuit mesh composed of , , _x, _y

L C C C used to model a two-level atom has only one resonance frequency representing the energy of the atomic excited level, and hence the current as well as the power absorption (losses)in this circuit will be maximal when the applied voltage source is on resonance. If, however, the two loops on the two sides are closed,there are two possible pathways to be excited, namely, the analog of the transitions |1  | 3 | 2 and |1 | 3 | 2 . This, however, mightcancel the two-level resonance, and the multilevel EIT results.

Now we shall investigate the analog to the induced transparency by analyzing the frequency dependence of the whole consumed power^Pp

 

^p of RLC circuit (defined as

      

^*



p p p p p p

1Re

P  2 V  I  ) supplied by the voltage source^Vp

 

^p to the resonant circuitR LC . Here, the equivalent capacitor _i C is the series combination of capacitors_i

( ):

1 1

1 2 1

1 1

, ^y , ^{x y}

x

e e e

x y x y x y

C C CC C

C C C C C

C C C C CC CC C C

  

    (5-8)

Ci

110

If we set L₁L₂ L and employ the definition of the currents

 

^{dq t}

 

i t  dt ,

 

1

 

1

i t dq t

 dt and

 

2

 

2

i t dq t

 dt we would have the following coupled differential equation set for the charges^{q t ,}

 

q t , 1

 

q t : 2

 

2

p

2 2 2

1 2

2 x y

d q dq V

q q q

dt  dt       L (5-9)

2

2 2

1 1

1 1 1

2 _x 0

d q dq

q q

dt  dt     (5-10)

2

2 2

2 2

2 2 2

2 _y 0

d q dq

q q

dt  dt     (5-11)

Here, the relevant parameters of frequency and decay (loss) are defined as R L/ ,

i Ri /Li

  ,_i² 1 /



L C_i e_i

 

i1, 2



,² 1 / LC



e



, ²_x 1 /



L C1 _x



and ²_y 1 /



L C2 _y



. Once the current^{i t}

 

is obtained, the power ^Pp

 

^p as a function of the frequency

p of the applied voltage source is given by

   

_{ }

   

2

p p p rms

p p 2 2

p p p p

R V

P R X

 

 

  (5-11)

with V_{p peak }1 V, _{p rms } ^p 0.707 V 2

V  V 

   

   

 

2 2

1 p 2 p

p p 2 2

2 2

1 p 1 p e1 2 p 2 p e2

/ /

1 / 1 /

x y

R C R C

R R

R L C R L C

 

 ^{ } ^   ^ ^ ^   ^

(5-12)

       

 

2

p p 1 p e1

p p p p e 2

2

1 p 1 p e1

1 / 1 /

1 /

1 /

Cx L C

X L C

R L C

  

  

 

    

 

 

 

      

   

 

2

p p 2 p e2

2 2

2 p 2 p e2

1 / 1 /

1 /

Cy L C

R L C

  

 

    

 

 

 

  

(5-13) ( )

i t

111

On the other hand, when the two loops on the two sides are open, and this implies that the two impedanceZ₁ andZ₂  , we would have

 

^{ }

 

2 p rms

p p 2

2

p 1 / p e

RV P

R L C

 ^    

(5-14)

In order to verify the above results, we shall simplify the circuit as Fig. 10a, Fig.10b and Fig.10c

(a) (b) (c)

5.10 The EIT-like RLC circuit can be simplified as a single closed loop with

different impedance included (a)Z_total  R jX ,(b) Ztotal



RR1

 

 j XX1



, (c) Ztotal



RR1R2

 

 j X X1X2



At first, we shall calculate the impedance, Z_total  R jX , of the simple loop on the middle of the circuit.(See Fig. 5.10a)

total p

p

Z R j L 1 R jX

 C



 

      (5-15)

where, X pL1 /

 

pC

Second, the equivalent impedance of loop on left side of the circuit can be written as below.

(See Fig.5.10b).

 

  ^{ }

   

1 p 1 p 1 p

1 1 1

1 p 1 p 1 p

1 / /

1 / 1 /

x

x

R j L C j C

Z R jX

R j L C C

  

  

   

       

  

 

    

(5-16)

where,

 

 

2

1 p

1 2

2

1 p 1 p e1

/ 1 / R Cx

R

R L C



 

    

(5-17)

112

   

 

2

p 1 p e1 p

1 2

2 p

1 p 1 p e1

1 / / 1

1 /

x

x

L C C

X R L C C

  

  

  

 

   

 

  

(5-18)

Third, the equivalent impedance of loop on left side of the circuit can be written as below.

(See Fig.5.10c).

 

  ^{ }

   

2 p 2 p 2 p

2 2 2

1 p 2 p 2 p

1 / /

1 / 1 /

y

y

R j L C j C

Z R jX

R j L C C

  

  

   

       

  

 

    

(5-19)

where,

 

 

2

2 p

2 2

2

2 p 1 p e2

/ 1 / R Cy

R

R L C



 

    

(5-20)

   

 

2

p 2 p e2 p

2 2

2 p

2 p 2 p e2

1 / / 1

1 /

y

y

L C C

X R L C C

  

  

  

 

   

 

  

(5-21)

Combine all these three impedances in series together, the total impedance of this circuit is

given by . The active and reactive power

sent to the RLC circuitareP_{r  }^V_{p rms  }I_prms^_cosand P_{i  }^V_{p rms  }I_prms^_sinrespectively, where, ^{T T} _{p rms } ^{p rms  p rms }

2 2 2 2

T T T T T T

cos R R , V V

Z R X I Z R X

    

 

The effective power is thus can be expressed in impedance as

   

 

p rms T T 2

r p rms 2 2 2 2 2 2 p rms

T T

T T T T

V R R

P V V

R X

R X R X

 

  

  

 

  (5-22)

The above two parameters can also be alternatively written as

     

4 4 1 2

T p 2 2 2 2 2 2 2 2 2 2

1 p p 1 2 p p 2

x y

R  L   

       

   

 

  

     

 

(5-23)

     

   

 

4 2 2 4 2 2

p 1 p 2

2 2

T p p 0 2 2 2 2 2 2 2 2 2 2

p 1 p p 1 2 p p 2

x y

X L    

  

        

     

 

   

     

 

(5-24)

Note that the collisional dephasing rates (shown in Fig. 1) depend on the pressure

' ' ' ' ' '

T T T ( 1 2) ( 1 2)

Z R  jX  R R R  j X  X  X

2, 2'

 

113

of the atomic vapor, and they would almost vanish in a dilute atomic vapor. Thus, the resistancesR and₁ R in our numerical simulation can be quite small (this corresponds to the ₂ fact that the collisional dephasing rates ₂, _2in a dilute atomic vapor are negligibly small).

IfR and ₁ R in the circuit vanish simultaneously , i.e.,₂ ₁ ₂ 0 one can have

 

T p minimum

R  L  R (5-25)

       

4 4

2 2

T p p 0 2 2 2 2

p p 1 p 2

maximum

x y

X  L  

    

   

     

 

 

  (5-26)

It follows that the active power will have two responding dip at frequency _p ₁and

p 2

  , which correspond to the two transparency windows in the four-level tripod-configuration atomic EIT.

(a)

(b)

114

5.6 Shift Parameters Result in Different Autler-Townes Triplet

According to the theoretical result of the consumed power equations represented in the preceding section, we plot the curves of the frequency-dependent active power in the circuit analog of the two-, three- and four-level systems in Fig. 5.11(a), (b), (c), respectively. The relevant parameters of the electronic components in the circuit are given in Table 1.

In order to confirm the above numerical result obtained by Matlab software, we shall now use ADS(Advanced Design System, Agilent software) to simulate the efficient power in the circuits with one loop, two loops and three loops, which represent the two-, three- and four-level atomic systems, respectively. The ADS result is presented in Fig. 5.12(a),(b),(c).

It can be seen that the Matlab result is almost exactly the same as the ADS result.

(c)

Fig. 5.11 The Matlab results for power absorption in the circuit analog of two-level

atomicsystem(a), three-level system (b) and four-level system (c). The relevant parameters of the electronic component are given in Table 1. There are onresonant peak, two Autler-Townes peaks, and three Autler-Townes peaks in the circuit analog of two-, three-, and four-level systems, respectively.

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