Ultracold Rydberg atoms for efficient storage of terahertz frequency signals using electromagnetically induced transparency

Accepted Manuscript Ultracold Rydberg Atoms for Efficient Storage of Terahertz Frequency Signals using Electromagnetically Induced Transparency

Sumit Bhushan, Vikas S. Chauhan, Raghavan K. Easwaran

PII: DOI: Reference:

S0375-9601(18)31038-7 https://doi.org/10.1016/j.physleta.2018.10.006 PLA 25335

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Physics Letters A

Received date: Revised date: Accepted date:

10 May 2018 1 October 2018 9 October 2018

Please cite this article in press as: S. Bhushan et al., Ultracold Rydberg Atoms for Efficient Storage of Terahertz Frequency Signals using Electromagnetically Induced Transparency, Phys. Lett. A (2018), https://doi.org/10.1016/j.physleta.2018.10.006

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Highlights The highlights of our work presented in this letter are as follows: (a) first proposal of a Quantum Memory device for storage of THz frequency signals in ultracold atomic ensemble in a two dimensional magneto optical trap (b) first proposal to implement EIT with all the relevant energy levels within the Rydberg manifold (c) very high Optical Depth of ∼ 690 (d) very high Delay Band width Product of 9.5 (e) very high maximum storage efficiency of ∼ 99%. All of the above features make our scheme a preferably feasible choice for realizing QM in THz frequency domain for future applications.

Ultracold Rydberg Atoms for Efficient Storage of Terahertz Frequency Signals using Electromagnetically Induced Transparency Sumit Bhushan, Vikas S. Chauhan and Raghavan K Easwaran* Department of Physics, Indian Institute of Technology Patna, Bihta-801103, Patna, India. *Corresponding author: [email protected]



Abstract Quantum communication with terahertz (THz) frequency signals has many advantages like reduced attenuation and scintillation effects in certain atmospheric conditions along with very high level of data security. In this work, we propose a scheme to realize Quantum Memory (QM) for efficient storage of terahertz (THz) frequency signals using Electromagnetically Induced Transparency (EIT) in an ultracold atomic medium of 87Rb Rydberg atoms prepared in a Two Dimensional Magneto Optical Trap (2D-MOT). The uniqueness of our scheme lies in the choice of the energy levels involved in the EIT process, all three of which have been chosen to be the Rydberg levels (enabling signal beam to be in THz) in a lambda type arrangement. This first of its kind proposal reveals that atomic media are a potential candidate for devising QMs which can store THz frequency signals. We have estimated that the Optical Depth (OD) in our scheme can reach a very high value of 690, very high maximum obtainable storage efficiency (ɻ) of ~99%, the group velocity (vg) can be as low as 5.07 x 103 m/s, and the Delay Bandwidth Product (DBP) can be as high as 9.5. All of these estimates emphasize the feasibility of our scheme as a QM device for efficient storage of THz pulses. Keywords: Quantum Optics, Quantum memory, Rydberg atoms, Electromagnetically Induced Transparency

realize a Quantum Memory (QM) [4–6] to efficiently store these THz signals which has applications in future Quantum Communication (QC) networks [7–10]. Memory elements are vital components of any communication network. So far, very little research has been done to explore QM for THz signals [11,12]. In these works, the obtained Delay Bandwidth Product (DBP) [13], which is a measure of the number of signal pulses that can be stored by the memory element, has reached values of 0.3. We have shown in this paper that our system is capable of reaching very high values of DBP in comparison to the earlier works. Moreover, efficiency of storage is another parameter where our system is having a very high value (approaching unity). Our proposed scheme is based on a lambda type Electromagnetically Induced Transparency (EIT) [14–16] in an ultracold atomic ensemble of 87Rb Rydberg atoms in a Two Dimensional Mangeto Optical Trap (2D-MOT) [17–19]. Our model is unique in the sense that both the probe (or signal) and control transitions are chosen to be in the Rydberg manifold so that the associated atomic transitions are in the THz domain, whereas the

1. INTRODUCTION Terahertz (THz) frequency signals (0.1 THz to 10 THz) experience reduced attenuation and scintillation effects (caused by atmospheric turbulence and fluctuations in humidity) in atmospheric conditions such as fog, smoke, and by airborne particles [1–3] in comparison to the IR frequencies. THz waves support very high bandwidth in comparison to the microwave and millimeter signals. Moreover, their high level of directionality reduces the probability of eavesdropping, thus ensuring secure communication. Although THz signals suffer from high attenuation while propagating through clouds but for long distance communications (such as space communication), these attenuations can be neglected [2]. These advantages motivate us to explore the field of THz communication. Quantum communication (QC) networks provide absolutely secure transmission of signal over remote locations. THz signals have not been explored from the point of view of quantum communications. In this letter, we are proposing a scheme to 1

current Rydberg EIT schemes involve mostly ladder type three level atomic media in which the upper transition is from an excited state to a Rydberg state [20,21]. The group velocity vg of the probe passing through a medium under EIT reduces sharply [22,23], enabling the medium to store the probe pulse and then release it on demand [24]. The choice of 2D-MOT systems to work with has the following advantages (a) a very high Optical Depth (OD) [17,25] which is the single most important factor [26] responsible for high efficiency of storage in a QM, (b) adjustable sample length L which is a key factor for optimizing the efficiency, and (c) possibility of obtaining high values of L. In the following paragraphs we present a detailed description of our proposed scheme and the estimates of values of various performance parameters like group velocity, OD, optimal efficiency, and DBP.

where, f is the oscillator strength for the considered transition, ʆ is the transition frequency, W is the value of Wigner 3j symbol, E is the electric field applied, and t is the pulse duration of the excitation laser. 

2. THEORETICAL MODELING  In our proposed scheme as shown in Fig.1, the atoms in the ground state 5S1/2 of 87Rb in a 2D- MOT [25] are excited to the Rydberg level 25P1/2 by a laser beam of 297 nm wavelength derived by frequency doubling a dye laser as explained in ref. [27]. The cooling and trapping of 87Rb atoms in a 2D-MOT involves transitions between 5S1/2 and 5P3/2 states and hence, if the 297nm laser pulses are sent to the excitation volume while the cooling and trapping process is still going on, efficient Rydberg excitation will not take place. Thus, the cooling and trapping lasers need to be switched off before the Rydberg excitation laser is switched on. The EIT has to be performed during the period ȴtoff for which the cooling and trapping lasers remain switched off. However, ȴtoff is decided on the basis of two factors (i) it should be lesser than the coherence time between the states ȁͳۧ and ȁʹۧ (because, after the coherence time, EIT would not sustain) and (ii) the atomic cloud’s expansion during this period should be negligibly small in comparison to the cloud size itself. The coherence time (= 1/ߛୢୣୡ where ߛୢୣୡ is the decoherence rate betweenȁͳۧandȁʹۧ. The calculation of decoherence rate has been presented later in this section.) is approximately 66μs. Thus, if the cooling and trapping lasers are switched off for say, 40μs (= ȴtoff), it can be shown that the expansion in the atomic cloud considered here [25] , would be ~ 3μm (calculated at a typical 2D-MOT temperature of 70μK [28])which is negligible in comparison to the atomic cloud size (10mm x 80mm cylinder) itself. It can be noted that during the period ȴtoff, the magnetic field coils are kept on as, in a 2D-MOT, the atoms are trapped along a line where the net magnetic field of these coils is zero. So, for ȴtoff = 40μs, the 297nm laser would see most of the trapped atoms in their ground states and hence excite them to the Rydberg state ȁͳۧ, and then EIT can be performed successfully as explained in the following paragraphs.



Fig. 1. Energy level diagram of the proposed EIT scheme. Atoms initially in the ground state 5S1/2 are excited to level ȁͳۧ by very strong 297 nm wavelength laser. ɏc, ɏp , and ɏg1 are the Rabi frequencies of control, probe and Rydberg excitation lasers respectively with. ɷc and ɷp, as the detunings of control and probe lasers from their respective transitions.

We have calculated the value of oscillator strength f for n = 25 to be 2.38 x 10-6 from the formula given in Ref. [29]. The value of Wigner 3j symbol W for the Rydberg transition 5S1/2 (J =1/2 , m = 1/2) to 25P1/2 (J =1/2 , m = 1/2) is written as [29] 

§1 / 2 1 1 / 2 · W =¨ ¸ (2) © 1 / 2 0 -1 / 2 ¹ After solving the expression given in eq. (2), we find W = 0.41. In our model, we want most of the atoms in the level ȁ‰ۧ to be excited to ȁͳۧ. If this happens, the energy level ȁ݃ۧ does not enter into the picture as far as EIT is concerned. The analysis then reduces to a common three level lambda type atomic system involving the energy levels ȁͳۧ, ȁʹۧ, and ȁ͵ۧ. Such a situation can be reached if the probability of transition ȁ‰ۧ - ȁͳۧ, given by eq. (1) is maximum, i.e. 1. Moreover, once the atoms are excited toȁͳۧ, they should remain there for a long time which means that the effects which lead to the decay from ȁͳۧ to other lower states should be negligible. In the following paragraphs, we present our method and argument of ensuring that the above happens.

The energy gap between the statesȁͳۧ and ȁʹۧ in Fig. 1 is ~8GHz and the linewidth used for the Rydberg excitation is ~ 700 kHz [27]. Thus, the laser tuned to ȁ‰ۧ - ȁͳۧ transition, does not excite the atoms to ȁʹۧ. There is a probability associated with this transition which is given by [29] 

§ P = sin2 ¨ ¨ ©

f

From eq. (1), it can be easily found that the probability of excitation reaches unity at a time t ~ 6.6ns if the excitation electric field E is 105 V/m (~13MW/m2). Thus, we excite the atomic sample to Rydberg state ȁͳۧ with pulses of duration 6ns. It is to be noted that even with an excitation probability of 1 with the pulse of above mentioned duration, there may be decay from ȁͳۧ due to  processes such as spontaneous emission which start in a time of the order of microseconds [21] in the present case. Moreover,

6 =e 2 E t · (1) W2 ¸ me 2πν = 2 ¸¹ 2

there are other mechanisms like dephasing due to dipole-dipole exchange and superradiant emission, which are together called collective decay or decay due to cooperative effects. This collective decay greatly depopulate Rydberg states [21,30]. The rates at which Rydberg states decay due to dipole-dipole exchange processes and superradiance are given as [31]

γ DD =

π d 2 NR 4ε 0 =

dg1E/Ŝ where dg1 is the transition dipole moment of the above transition. To calculate dg1 we have used the corresponding formula given in ref. [35] and we found dg1 ~ 1.85 x 10-31 C.m. With all these values, we estimate Nb ~ 1.92 x 10-6 and Rb ~ 1.35ʅm. These values are very small and thus the effects of Rydberg blockade can be neglected [36] in our model. Thus, it is seen that with carefully chosen energy levels such as the one in our model, and with applying appropriately estimated value of the excitation laser intensity and pulse repetition rate, almost all the atoms from the ground state ȁ‰ۧ can be excited to the Rydberg level ȁͳۧ. This Rydberg level can be treated as the new ground state for the purpose of EIT. Hence, we can take NR = NA = 1011 atoms/m3. The rest of this section provides an explanation of the mechanism by which EIT is feasible in our model.



(3) ƒ†

π d 2 NR L γS = (4) 3ε 0 =λ

For the EIT to be feasible, we have to fulfill the following conditions: (a) Both probe and control transitions should be dipole allowed. The selection rule for dipole allowed transition is ο۸ ൌ ૙ǡ േ૚, where ۸ is the total angular momentum of a given energy level. For the energy levels ȁ૚ۧ, ȁ૛ۧ, and ȁ૜ۧ, the values of ۸ are 1/2, 3/2, and 3/2 respectively. Hence, for both probe and coupling transitions, this condition is fulfilled.

where, ߛୈୈ is the decay rate due to dipole-dipole exchange, ߛୗ is the decay rate due to superradiance, d is the transition dipole moment for Rydberg-Rydberg transition, NR is the atomic density in the Rydberg state, L is the length of the atomic cloud (8cm in our system), and ʄ is the wavelength of the Rydberg-Rydberg transition. Now, it can be calculated that the decay times due to dipole-dipole and superradiance emission are of the order of millisecond and microsecond respectively for an atomic density of the order of 1011 atoms/m3. In particular, the dipole-dipole induced decay time for a transition to 25S state is 10-1 ms and the superradiance induced decay time is of the order of 10-1 μs. Thus, we see that decay due to superradiance is dominant. Thus, eventhough the lifetime of 25P3/2 state is 27μs [32] , the decay from it starts much earlier and hence, the repetition rate for Rydberg excitation pulses should be much lesser than microsecond so that the population in this state is maintained. In our system, the repetition rate for the 297nm laser pulses should be a few nanoseconds.•ƒƒ––‡”‘ˆˆƒ…–ǡ…‘‡”…‹ƒŽŽƒ•‡”•‘ˆ ƒ‘•‡…‘† ȋ œȌ ”‡’‡–‹–‹‘ ”ƒ–‡ ƒ”‡ ƒ˜ƒ‹Žƒ„Ž‡Ǥ Š‡ ‘–Š‡” …‘ŽŽ‡…–‹˜‡ †‡…ƒ›‡…Šƒ‹••ƒ”‡‡‰Ž‹‰‹„Ž›•ƒŽŽ„‡…ƒ—•‡ɉ‡–‹‘‡†ƒ„‘˜‡‹• —…ŠŽ‡••‡”–Šƒ–Š‡…Ž‘—†•‹œ‡ȏʹͳȐǤ

(b) The lasers for probe and control transitions should be readily available. The frequencies corresponding to probe and control transitions in our system are 0.705 THz and 0.696 THz respectively and these frequencies can be derived from Quantum Cascade Lasers (QCL) [37–39]. (c) Linewidth of the lasers should be lesser than those of the transitions they are addressing. This makes sure that the nearby Rydberg levels are not excited and only the intended transitions are addressed. Linewidth of the probe and coupling transitions used in our model are both of nearly 12kHz. The THz QCLs given in ref. [39] are less than 300 Hz and can be good choice for the experimental realization of our model. (d)For observing the EIT signal, ષ‫ ب ܋‬ඥࢽ‫[ ܋܍܌ࢽ ܘ‬16] where ࢽ‫ ܘ‬is the decay rate of the probe transition and ࢽ‫ ܋܍܌‬is the decoherence between ȁ૚ۧ and ȁ૛ۧ. The lifetimes of the Rydberg levels ȁ૛ۧ and ȁ૜ۧ are given by 29.15ʅs and 13.995ʅs respectively [32] and hence their decay rates are ࢽ૛ = 0.03 MHz and ࢽ૜ = 0.07 MHz. The lifetimes given above have been calculated by the formula [32]

One more phenomenon which affects the Rydberg state population is the so called dipole blockade effect [33] which prevents all the atoms from being excited to Rydberg energy levels. As a consequence of the dipole blockade, each atom that is Rydberg excited, blocks a number Nb of atoms from being excited to the Rydberg level by creating a blockade sphere of radius Rb. Thus, even with the transition probability of 1, we would have a reduced number of atoms in the Rydberg level 25P1/2. The value of Nb is given by [33] 

ª§ 4π N b ~ «¨ ¬© 3

º · ¸ N A C6 =Ω g 1 » ¹ ¼

1

τ effective

45

(5)

 where, NA is the initial number density of atoms (prepared in a 2DMOT, as mentioned above) in the ground state. The value of NA that we have taken is equal to 1011 atoms/m3. The value of C6 parameter [34] for 25P1/2 is -1.59 x 10-64 Hz.m6 and the Rabi frequency ɏg1 of the transition |‰ۧ - |ͳۧ is calculated by using ёg1 = 3

=

1

τ0

+

1

τ BBR

(6)

where ࣎‫ ܍ܞܑܜ܋܍܎܎܍‬is the effective lifetime of the Rydberg state, ࣎૙ is the radiative decay (spontaneous emission), and ࣎۰۰‫ ܀‬is the decay due to black body radiation from the Rydberg state. So, the values of lifetimes taken here contain these effects and hence decay caused by them need not be considered separately. Now, as explained above, since the decay rate ࢽ૚ of ȁ૚ۧ is negligible. Thus, ࢽ࢖ (= ሺࢽ૜ ൅ ࢽ૚ ሻΤ૛) and ࢽ‫=( ܋܍܌‬ሺࢽ૛ ൅ ࢽ૚ ሻΤ૛ሻ are estimated to be

0.035 MHz and 0.015 MHz. Thus, we take ષ‫ ܋‬to be 1.75 MHz which is ~ 50ࢽ࢖ ‫ ب‬ඥࢽ‫( ܋܍܌ࢽ ܘ‬ൎ 0.02 MHz).

(b)

(e) The probe and control laser beams may experience an increase in beam waist due to divergence. The net beam waist after considering divergence should not exceed the dimensions of the cloud. This ensures that majority of atoms interact with these laser beams and hence the losses are minimized. Now, the cross section of the 2D-MOT considered for this work is 10mm [25] while the size of the beam waist of the probe beam because of divergence is of the order of 1mm [40]. Thus, the probe beam is spatially contained within the 2D-MOT and hence interacts with maximum number of atoms.

 Fig. 2. Top(a) to bottom(b): Real and imaginary part of the susceptibility as a function of probe detuning in our proposed scheme.

Hence, in our model experimental realization of Rydberg EIT with all the relevant energy levels within the Rydberg manifold should be feasible in a 2D-MOT as mentioned. In order to observe EIT, we have to calculate the susceptibility of the four level atomic system shown in Fig. 1.The Hamiltonian for the four level atomic system shown in Fig. (1) can be written as

ª −2ω p « = « −Ω g1 H= « 2 0 « «¬ 0

−Ω g1 0

0 0

0

2 (δ c − δ p )

−Ω p

−Ωc

0 º −Ω p »» −Ωc » » −2δ p »¼

From Fig.2(a), it can be seen that the dispersion is very steep around the probe beam resonance which leads to a very sharp reduction in the group velocity of the probe pulse.

3. ESTIMATION OF GROUP VELOCITY, DBP, AND STORAGE EFFICIENCY The group velocity of the probe pulse in an EIT medium is given by [25]

(7)

vg = (9)

where, ɷc = ʘc – ʘ32 and ɷp = ʘp – ʘ31 are the detunings of control and probe lasers respectively from their corresponding resonant transitions. ʘc (ʘp) and ʘ32 (ʘ31) are the frequency of the coupling (probe) laser and the resonance frequencies of their respective transitions. Under the assumptions ߗg1‫ ب‬ɏc, ɏc ‫ ب‬ɏp and ࢾc=0 the susceptibility for the probe transition is given as (we have also assumed zero detuning for the 297nm excitation)

χ=

4 N R d p2 § δ p − γ dec ¨ ε 0 = ¨ 4 ( iδ p + γ dec )( iδ p + γ p ) + Ω c2 ©

· ¸ ¸ ¹

Ωc2 L  2γ p .( OD )

where L is the medium length. The value of OD in an EIT medium is given by [19] OD= NRʍL where ߪ ൌ ͵ߣଶ୮ Τʹߨ is the absorption cross section and is estimated at 8.8 x 10-8 m2 for the probe transition. Using the estimated values of the relevant parameters, we calculate the OD of our medium to be ~690 which is very high in comparison to other Rydberg EIT schemes [41] . We have plotted the variation of vg as a function of ɏc for different values of the medium length L as shown in Fig.3. From Fig.3, we find that the lowest value of vg obtainable in our scheme [25] for L= 8cm is ~5.07 x 103 m/s, leading to a large group delay of 15.8μs. At such small group velocities, the probe pulse can easily be stored inside the EIT medium by switching off the control field once the pulse is fully inside the medium. For adiabatic storage [42], the group delay ο‫ݐ‬ୢୣ୪ୟ୷ should be smaller than ͳΤߛୢୣୡ . This condition puts an upper limit on the obtainable group delay in an EIT medium. For our system, this condition is satisfied and hence adiabatic storage of quantum information is feasible. For efficient storage, it is necessary that the pulse is contained fully inside the medium both spatially and spectrally. These conditions can be fulfilled if (a) ȴfEIT >> ȴfp and (b) vgȴtp << L where, ȴfEIT (ȴfp) is the bandwidth of EIT (probe pulse), and ȴtp is the time duration of probe pulse. Moreover, for the complete transfer of the quantum information contained in the probe pulse to the atomic states before the pulse gets absorbed by the EIT medium, it is necessary that [43] ȴtpߛୢୣୡ >> 1. The value of ȴfEIT for our system is 6.6MHz. Thus, for complete spectral confinement of the probe pulse within the EIT window, we take the bandwidth of it as ȴfp = 0.6MHz which corresponds to ȴtp = 1.6μs. With these values, all of the conditions for efficient storage mentioned above are satisfied. The maximum storage efficiency is given by [44] Ʉ ൌ ͳ െ ͷǤͺΤ and thus, for

(8)

where, dp is the transition dipole moment of the probe transition. In order to see the variation of susceptibility of the medium in Fig.1, we substitute the values of NR, ߛ୮ , and ߛୢୣୡ as obtained above and ʄp ~ 422ʅm, dp ~6.35 x 10-25 C.m (calculated by the method as mentioned in the description following eq. (5)), and ɏc = 50ߛ୮ (corresponding power ~ 0.05pW), along with standard values of other parameters in eq. (8) to obtain the plot shown in Fig.2.  (a)

 4

Rydberg level) energy levels in the Rydberg manifold, (c) very high OD of ~ 690 (d) the lowest group velocity achievable is ~ 5.07 x 103 m/s, (e) high DBP of 9.5 in comparison to other slow light systems for THz frequencies, and (f) very high maximum storage efficiency of ~99%. All of the above features make our scheme a preferably feasible choice for realizing QM in THz frequency domain for future applications. It should be noted that further studies are required for quantum information applications, in particular regarding impact of the laser (297 nm) phase imprinted at each pulses and the magnetic field gradient in the 2D-MOT which induces a magnetic dispersion of few Gauss over the cloud size leading to a Zeeman dephasing.

our scheme the storage efficiency can go to the maximum value of ~99%. Another very important parameter which measures the capability of a quantum memory is the DBP [26] which is the product of the group delay (L/vg) and the bandwidth of the probe pulse (1/ȴtp) and it represents the number of pulses that can be stored by the QM [45,46]. The variation of DBP as a function of the group velocity for different values of medium lengths L is as shown in Fig.4. 

ACKNOWLEDGEMENT We thank Department of Science and Technology, Government of India and Ministry of Human Resources Development, Government of India for the financial support. 



REFERENCES

FIG. 3. Group velocity as a function of the Rabi frequency of the control beam for different values of the medium length L.

[1]

 It can be seen from Fig.4 that the value of DBP increases with the medium length L. For the storage of 0.6MHz pulse mentioned above, the DBP is 9.5, which is far higher than that in other slow light systems for THz frequencies such as metamaterials [11,12]. For example, the DBP obtained in ref. [11] is 0.2 only. 

[2] [3]

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[5]



FIG.4. DBP as a function of the group velocity obtainable for different values of the medium length L.

[6] The high DBP and storage efficiency and very small decoherence rate obtainable in our scheme suggest that ultracold ensemble based QMs are preferable choice for THz frequency signal storage for future quantum communication network at these frequencies.

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4. CONCLUSION

[8]

To conclude, we have proposed a scheme for highly efficient storage of quantum information at THz frequencies which has direct application in future quantum communication networks at these frequencies. The highlights of our scheme are: (a) first proposal of a QM device for storage of THz frequency signals in ultracold atomic ensemble, (b) first proposal to implement EIT with all the relevant (i.e., the three levels associated with the EIT process,(ȁ܏ۧ is involved only in the excitation of atoms to the

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