Highly sensitive atomic based MW interferometry

We theoretically study a scheme to develop an atomic based micro-wave (MW) interferometry using the Rydberg states in Rb. Unlike the traditional MW interferometry, this scheme is not based upon the electrical circuits, hence the sensitivity of the phase and the amplitude/strength of the MW field is not limited by the Nyquist thermal noise. Further, this system has great advantage due to its much higher frequency range in comparision to the electrical circuit, ranging from radio frequency (RF), MW to terahertz regime. In addition, this is two orders of magnitude more sensitive to field strength as compared to the prior demonstrations on the MW electrometry using the Rydberg atomic states. Further, previously studied atomic systems are only sensitive to the field strength but not to the phase and hence this scheme provides a great opportunity to characterize the MW completely including the propagation direction and the wavefront. The atomic based MW interferometry is based upon a six-level loopy ladder system involving the Rydberg states in which two sub-systems interfere constructively or destructively depending upon the phase between the MW electric fields closing the loop. This work opens up a new field i.e. atomic based MW interferometry replacing the conventional electrical circuit in much superior fashion.

explore a six-level loopy ladder system which replaces the traditional electrical circuits based MW interferometry by the atomic MW interferometry, as the absorption property of the probe laser has phase dependency on the MW fields. This is based upon the interference between two sub-systems driven by the MW fields forming the loop. The limitation of the atomic based MW interferometry is again same as in case of the atomic based MW sensor studied with four-level system 6,8 and is not limited by the thermal noise. But this system is two orders of magnitude more sensitive to field strength (upto 80 nV/cm) in comparison to the previously explored system 6,8 due to its loopy nature. There are loopy system which has been studied previously and has phase sensitivity but loop is completed using the weak magnetic dipole transition 22 . In contrast to the previous system this six-level loopy system involves allowed electric dipole transition. This paper is organized as follows. In the section namely "Method", we describe the method of realizing the six-level loopy ladder system in Rb and possible experimental set-up. In subsequent sub-section we present the semi-classical model and solution for the relevant density matrix element. Further we provide the physical interpretation of the obtained mathematical solution in terms of the interference between the two sub-systems and in terms of the dressed state picture. In the next section namely "Results" we present various results including the lineshape of the probe absorption, the phase dependency of it, the comparison of the amplitude/strength sensitivity of this system with the previously studied four-level system and the frequency range. Finally in the section namely "Discussion" we give our conclusion for this study.

Method
Realization of the system. The considered six-level loopy ladder system is shown in Fig. 1a. The probe laser at 780 nm is at the D 2 line i.e. driving the 5 S 1/2 → 5 P 3/2 transition in the Rb. The control laser at 480 nm is driving the → 5P n S . The n ryd 1 , n ryd 2 , n ryd 3 and n ryd 4 are rydberg states which are chosen according to the frequency range of the MW field. The typical experimental setup for phase dependent MW electrometry is shown in Fig. 1(c) in which a probe laser at 780 nm and a control laser at 480 nm are counter-propagating inside the Rb cell. The four MW control fields are generated by a single frequency synthesizer having arrangements of controlling the frequency, phase and the amplitude or the four different MW field frequencies combined using a frequency combiner (e.g. ZN4PD-02183-S+ from minicircuit company can be operated between 2-18 GHz). The output of the frequency synthesizer or combiner is amplified and fed to MW horn. All four MW fields are propagating perpendicular to the probe and the control lasers with a uniform phase inside the Rb cell.
Semi-classical analysis. The electric field, associated with the transition |i〉 → |j〉 is where E ij is amplitude, ω ij is the frequency and φ ij is the phase. We define Rabi frequency Ω = φ d E e / ij ij ij i ij  for the transition |i〉 → |j〉 having the dipole moment matrix element d ij . Please note that Ω ij is a complex quantity which can be written as |Ω ij | φ e i ij , where φ ij is due to the phase of the electric field associated with it. The Rabi frequencies of the probe and the control lasers are Ω 12 and Ω 23  In general, the Hamilitonian H is time dependent except for a particular condition when δ 34 − δ 45 − δ 56 + δ 36 = 0. The time evolution of the density matrix, ρ is given by Linblad master equation as where, L[ρ(t)] is Linblad matrix and defined as below. Where, Γ ij is the decay of the population from state |i〉 (i = 1, 2, .. to 6) to state |j〉 (j = 1, 2, .. 6) and Γ i is the total population decay rate of state |i〉. In the case of the weak probe, the population transfer does not take place and it is completely irrelevant to know the population dynamics between different levels. The only important parameter is Γ i and Γ j , i.e. the total decay rate of states, which governs the decoherence rate (γ ij dec ) between the two levels |i〉 , which includes natural radiative decay of excited state, Γ 2 = 2π × 6 MHz and the 780 nm laser linewidth of 2π × 50 kHz. We also take γ γ γ γ γ π = = = = = × 2 100 kHz dec d ec dec d ec dec 13 14 15 16 mainly dominated by the laser linewidths of 780 nm and the 480 nm as compared to the radiative decay rate (=2π × 1 kHz) of the Rydberg states |3〉, |4〉, |5〉 and |6〉 7 . We also take γ dec = 2π × 500 kHz in some cases in order to check it's stringency.
From Eqs 3, 4 and 5 we get 36 coupled differential equations with the property ρ ρ = ⁎ ij ji . In order to solve these set of coupled equation we adapt similar method as in the case of previously studied multi-level systems 23 .
In the case of weak probe approximation, there will be no population transfer and hence the time evolution of the population i.e. the diagonal terms of the density matrix such as ρ 11 , ρ 22 , ρ 33 , ρ 44 , ρ 55 , and ρ 66 can be ignored. Similarly the time evolution of the off-diagonal terms ρ ij for i = 2; j = 3, 4, 5, 6 and i = 3; j = 4, 5, 6 and i = 4; j = 5, 6 and i = 5; j = 6 can be also ignored. The time evolution of the relevant density matrix element is given below.    dec   dec   dec   13  13  12  23   14  14  12  23  34   15  15  12  23  34  45   16  16  12  23  34  45  56 Now, we apply the four-photon resonance condition for the MW fields i.e. δ 34 − δ 45 − δ 56 + δ 36 = 0. In this case the system will reach steady state i.e. ρ =  0 ij , for all the elements on the time scale of few tens of 1/Γ 2 as shown in Fig. 2. In the weak probe condition and in the steady state, ρ 11 ≈ 1, ρ 22 ≈ ρ 33 ≈ ρ 44 ≈ ρ 55 ≈ ρ 66 ≈ 0 and ρ ij = ρ ji ≈ 0 for i = 2; j = 3, 4, 5, 6 and i = 3; j = 4, 5, 6 and i = 4; j = 5, 6 and i = 5; j = 6. Finally, we get the following set of equations  The refractive index, n of the probe laser is related with the density matrix element, ρ 12 as is the wavelength of the probe laser and N is atomic number density 24,25 . The imaginary part of n is related with the absorption and real part with dispersion. We define the normalized absorption [(Γ 2 /Ω 12 ) Im(ρ 12 )] i.e. for the stationary atoms, the absorption of the probe laser at resonance in the absence of all the control lasers is 1.
In order to verify the approximation made above, we have checked the analytical solution of ρ 12 given by the Eq. 8 and the complete numerical solution in the steady state for various values of control fields and detunings. It has excellent agreement between complete numerical and approximated analytical solution as shown in Fig. 3. The solution for ρ 12 in Eq. 8 has the following interpretation.
Interpretation. Interference between two sub-system. Equation 8 looks very complicated but it can be interpreted in the following simple way. The closed loop system can be realized by two open loop sub-systems |3〉 → |4〉 → |5〉 → |6〉 and |3〉 → |6〉 → |5〉 → |4〉 shown with red and green arrows respectively as shown in Fig. 1b. These two sub-system shares a common |1〉 → |2〉 → |3〉 ladder system. In order to understand the absorption property of the probe laser Ω 12 , we switch on the control fields one by one and in the sequence for the two  23 and expressed by EITATA1 in Eq. 8. (In order to understand the transparency and absorption in the sequence, we strongly advice the readers to see the paper 23 ). The other path shown with green color will also cause EITATA by sequence of the control fields Ω 36 unk , Ω 56 ref and Ω 45 ref which is expressed by EITATA2. Further, these two sub-system causing EITATA1 and EITATA2, interferes with each other and expressed by the Int term in the Eq. 8, which is phase(φ) dependent.
In the other words, the closed loop |3〉 → |4〉 → |5〉 → |6〉 → |3〉 causes absorption against EIT created by the control laser Ω 23 . The closed loop has two-open loop sub-systems which interfere destructively (for φ = 0) and constructively (for φ = π) with each other. As shown in Fig. 4a , there is a complete transparency at the line center for φ = 0. This is due to perfect destructive interference between the two-subsystems as the strength is same for both, i.e. EITATA1 = EITATA2. There is maxi-mum absorption at t he line center for φ = π as t he two sub-systems are inter fer ing const r uc t ively. For , there is a absorption peak at the line center for φ = 0, as shown in Fig. 4b. This is due to unequal strength of the individual system (EITATA1 > EITATA2), hence the destructive interference between them is not perfect.
Dressed state approach. At high Rabi frequencies (much greater than the absorption peaks linewidths) of the control lasers and MW fields, the linewidth of the absorption peak can be explained using dressed state picture. In this condition there is no interference between the absorption peaks as they are well separated from each other. The position of the absorption peak is determined by the eigenvalues of the Hamiltonian associated to the control fields as given below   which is phase dependent. In order to crosscheck the expression for the linewidth, we fit (shown with black solid line) the central peak of the normalized absorption obtained by Eq. 8 with Lorentzian profile to find the linewidth for three different phases as shown in Fig. 4. The fitted linewidths for φ = 0, φ = π/2 and φ = π are 0.13Γ 2 , 0.47Γ 2 and 0.64Γ 2 respectively, while the calculated linewidths are 0.13Γ 2 , 0.39Γ 2 and 0.54Γ 2 respectively. There is a small mismatch between the fitted and the calculated linewidths by the dressed state approach for φ = π/2 and φ = π. This is because, as we see in Fig. 4, the central absorption peak is broadened for φ = π/2 and φ = π and the interference between peaks starts playing a role in the modification of the linewidth similar to three level system 26 .

Results
Probe laser absorption. The normalized absorption (Im(ρ 12 )Γ 2 /Ω 12 ) vs probe detuning (δ 12 ) for three different phases, φ = 0,π/2 and π is shown in Fig. 4. For the central absorption peak i.e. at δ 12 = 0, only the linewidth depends upon the phase but not the position, while both the position and the linewidth depends upon the phase(φ) for the other four absorption peaks. This has been explained in the previous section. Now, we consider the effect of the temperature as lineshape of EIT is significantly changed by the thermal averaging [27][28][29][30][31][32] . The thermal averaging of ρ 12 is done numerically for the room temperature (T = 300 K) for the counter-propagating configuration of the probe (Ω 12 ) and the control laser (Ω 23 ) with wave-vectors k 780 and k 480 respectively by replacing δ 12 with δ 12 + k 780 v and δ 23 with δ 23 − k 480 v for moving atoms with velocity v, while the Doppler shift for the MW fields are ignored. Further the ρ 12 is weighted by the Maxwell Boltzman velocity distribution function and integrated over the velocity as , where k B is Boltzman constant and m is atomic mass of Rb. The integration is done over velocity range which is three times of k T m B . The Doppler averaging changes the absorption profile significantly as shown in Fig. 5. One of the interesting modification is the phase dependency of the probe laser absorption at the zero detunings of the probe. The probe laser absorption is minimum for φ = 0 and maximum for φ = π as shown with red and blue curve respectively in Fig. 5. This modification is due to mismatch of Doppler shift for probe at 780 nm and the control at 480 nm for moving atom. Please note that without thermal averaging at zero detunings of the probe, control laser and MW fields, probe laser absorption has no significant difference between φ = π/2 and π.
Phase sensitivity. Sinusoidal behavior. As seen in the previous section that the absorption profile of the probe laser depends upon the phase, φ. Please note that the previously studied (i.e. four-level) system [6][7][8][9][10][11] were insensitive to the phase of the MW field. This is also clear from Eq. 8 in the special case with Ω = Ω = Ω = 0 , which reduces the six-level loopy ladder system to four-level system and will have no phase dependency.
The probe absorption at room temperature vs the phase φ with all the detunings to be zero is shown in Fig. 6. From the plot shown with red open circle in Fig. 6a we observe more than 15% change in the probe absorption for the change of the phase from 0 to π for the chosen combinations of the control Rabi frequencies. In particular, we have chosen low value of Ω = . Γ  + θ), where A, B, f and θ are kept as free parameters that yields f = 1 and the fitting is shown with black curve in Fig. 6a. Now, choosing a high value of Ω = . Γ 2 5 36 unk 2 and keeping the other parameters unchanged, we observe more than 80% change in the probe absorption for the change of the phase from 0 to π as shown crossed red points, but there is a deviation from sinusoidal behavior. This deviation is compared with the fitted black curve as shown in Fig. 6b. On increasing the value of Ω 34 ref to 3Γ 2 and keeping the other parameters unchanged, there is a splitting of the absorption at φ = π as shown by the solid circled points in this figure. Optimization of sensitivity. Now, we maximize the phase sensitivity for this system for given value of Ω 36 unk by using the parameters, Ω 23  , which is a measure of the phase/strength sensitivity of the system and is to be maximized. For given value of Ω 36 unk , we maximize the S by minimizing 1/S or -S using matlab inbuilt function "fmincon" treating Ω 23  We first consider the case without thermal averaging i.e. T = 0. The maximized sensitivity, S max vs Ω 36 unk is plotted in Fig. 7(a) Fig. 7(b). The optimum value of the Ω 23 is as high as possible which is 5Γ 2 in this case as it is bounded by this limit. This is more clear from the Fig. 8 and Ω 36 unk . Next, we consider the room temperature case (T = 300 K), which makes the problem a bit more complicated, as the lineshape of the absorption gets modified significantly as described previously. The maximum sensitivity (S max ) vs Ω 36 unk is plotted in the Fig. 9(a). The S max at T = 300 K is much lower than the case at T = 0 as the saturation point is around Ω 36 unk = 1.5 Γ 2 as compared to 0.05Γ 2 and hence at T = 0 the system can detect the phase of lower values of Ω 36 unk . Unlike the case of T = 0, in this case for S max the value of Ω 23 ≠ 5Γ 2 but has optimum values as shown in Fig. 9(b).
Strength sensitivity. The quantity, S defined above can also be used as a measure of the strength/amplitude sensitivity for Ω 36 unk for the six-level loopy ladder system. Now we compare the strength sensitivity of the six-level loopy ladder system with the previously studied four-level system [6][7][8][9][10][11] . The solution of ρ 12 for the four-level system    The subscript (4l) indicates for four-level system. Further the thermal averaging can be done in a similar fashion as in the case of the six-level system i.e. . We define the strength sensitivity for the four-level system for unknown Ω 36 unk as change in the absorption in the presence and the absence of the . We maximize the sensitivity of the four-level system adapting similar method as for the six-level system but with only one optimizing parameter i.e. Ω 23 .
First, we consider T = 0 case. The maximized strength sensitivity for the six-level loopy ladder system and the four-level system is compared in Fig. 10. From this figure it is clear that the six-level system has more sensitivity as compared to the four-level system as shown in Fig. 10(a). In order to quantify this comparison, we plot the ratio of the sensitivities of the six-level to four-level system in Fig. 10(b). The ratio is more for the low values of the Ω 36 unk . The increased sensitivity for the six-level loopy system is due to the interferometric nature of the system where the effect of small Ω 36 unk is enhanced by the large values of the Ω 34 ref , Ω 45 ref and Ω 56 ref as the int term in Eq. 8 involves multiplication of these quantities. The strength sensitivity of both the systems decreases with increased γ dec (from 2π × 100 kHz to 2π × 500 kHz) but the effect is more for the four-level system in comparison to the six-level system as shown Fig. 10b. Now, we consider the case at the room temperature. The strength sensitivity for the six-level and previously studied four-level is plotted in Fig. 11(a). Form this plot it is clear that the six-level system has much superior strength sensitivity as compared to the four-level system. Further we quantify the comparison by plotting the ratio (R) of the sensitivities of the six-level to the four-level for different values of Ω 36 unk in Fig. 11(b). In order to check the stringency of γ dec on the sensitivity, we also plot S max for these two systems taking γ dec = 2π × 500 kHz.
We also plot the R vs maximum sensitivity (S max ) of the six-level system which gives the information about the possibility of the detection of Ω 36 unk . This is an important plot because there is a possibility that the R might be huge but can not be detected by the six-level system as well. The detection of S max up to 1% is very much feasible using locking detection. At this value of sensitivity for the six-level system, the sensitivity of the four-level system will be around 1 150 % as shown in Fig. 12. Finally one more important point is that, for the six-level loopy ladder system the MW field Ω 36 unk can be detected by just varying the phase of the reference MW fields, while in the case of the four-level system we need to insert and remove MW mechanical shield.

Discussion
In conclusion we theoretically study a six-level loopy ladder system using Rydberg states for the phase sensitive MW or RF electrometry. This is based upon the interference between the two sub-systems of EITATA. In counter-propagating configuration of the probe and control laser there is a change of the lineshape of the probe absorption due to Doppler averaging. The limitation of the proposed system is the decoherence rate between the ground state and the Rydberg states but not the thermal Nyquist noise as in the case of the electrical circuit based MW interferometry. The previously explored four-level atomic system has the same limitation and is already much superior than the electrical circuit for the strength sensitivity, frequency range and spatial resolution. This proposed system further improves the sensitivity by two orders of magnitude, removes the drawback of the phase insensitivity of the previous atomic four level-system and retains the advantages of the large frequency range of operation and spatial resolution. This system provides a great possibility to characterize the MW or RF electric fields completely including the propagation direction and the wavefront. This work will be quite useful for MW and RF engineering hence in the communications specially in active radar technologies and synthetic aperture radar interferometry. Figure 11. (a) S max (%) vs Ω Γ / 36 unk 2 for six-level loopy and four-level ladder system (b) ratio (R) of the sensitivity between six-level and four-level system vs Ω Γ / 36 unk 2 at T = 300 K with all the detunings to be zero and for γ dec = 2π × 100 kHz and γ dec = 2π × 500 kHz. Figure 12. Ratio (R) of the sensitivity between six-level and four-level system vs S max (%) of the six-level system at T = 300 K. The variation of S max (%) corresponds to range of Ω 36 unk from 0.005Γ 2 to 0.02Γ 2 .