Multi-mode Jaynes-Cummings model results for the collapse and the revival of the quantum Rabi oscillations in a lossy resonant cavity Najirul Islam and Shyamal Biswas

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Multi-mode Jaynes-Cummings model results for the collapse and the revival of the

quantum Rabi oscillations in a lossy resonant cavity

Najirul Islam and Shyamal Biswas∗

School of Physics, University of Hyderabad, C.R. Rao Road, Gachibowli, Hyderabad-500046, India

(Dated: February 7, 2022)

We have numerically obtained theoretical results for the collapse and the revival of the quantum

Rabi oscillations for low average number of coherent photons injected on a two-level system in a

lossy resonant cavity. We have adopted the multimode Jaynes-Cummings model for the same and

especially treated the “Ohmic” loss to the walls of the cavity, the leakage from the cavity, and the

loss due to the spontaneous emission through the open surface of the cavity. We have compared our

results with the experimental data obtained by Brune et al [Phys. Rev. Lett. 76, 1800 (1996)] in

this regard.

PACS numbers: 42.50.Pq (Cavity quantum electrodynamics; micromasers), 42.50.Ct (Quantum description

of interaction of light and matter; related experiments)

I. INTRODUCTION

Collapse and revival (CR) of the quantum Rabi oscil-

lations of a two-level system (atom/molecule) is an in-

teresting area of research in the ﬁeld of cavity quantum

electrodynamics [1–11]. Eberly et al ﬁrst predicted the

phenomenon of the CR within the single-mode Jaynes-

Cummings (J-C) model [12] for the quantum Rabi os-

cillations of a two-level system interacting with coherent

photons in a cavity [1]. The CR was subsequently ob-

served by investigating the dynamics of the interaction

of a single Rydberg atom with the resonant mode of an

electromagnetic ﬁeld in a superconducting cavity [5]. The

CR may ﬁnd applications in supersymmetric qubits [11].

While the existing theories [1–4,10,11] for the CR usu-

ally require a large average number of photons (¯n1) in

the coherent ﬁeld, a seminal experiment [7] on the same

was carried out by Brune et al for a low average num-

ber of photons (¯n&1) in the coherent ﬁeld. In fact,

as far as we know, all the experiments on the CR were

carried out for low average number of photons [5,7,13]

except the one [14] carried out for ¯n= 13.4. Hence we

theoretically investigate the CR for a low average num-

ber of photons in a coherent ﬁeld. Theory for the CR is

also available for low average number of injected coherent

photons as well as for all values of the average number

of the injected coherent photons [6,7,10,14–17]. This

theory takes only the resonant mode into account for the

light-matter interactions. We are, however, interested in

considering multi-modes into account.

J-C model takes only the resonant cavity mode into

account for the explanation of the CR of the quantum

Rabi oscillations of a two-level system in a loss-less cavity

[1,12]. However, the cavities are not loss-less in reality

[7]. This brings a frequency broadening as well as the ap-

∗Electronic address: sbsp [at] uohyd.ac.in

pearance of multi-modes around the resonant mode into

account. Brune et al ’s experiment on the CR were car-

ried out in a lossy resonant cavity of the mode quality

factor Q= 7 ×107[7]. The schematic diagram for the

two-level system interacting with the injected coherent

photons in the lossy resonant cavity is shown in ﬁgure

1. It is clear from ﬁgure 1how the injected coherent

photons are introduced into the cavity and how the two-

level system is interacting with the multi-modes of the

injected coherent photons in the cavity. Losses from the

cavity are shown by the wavy arrows in the same ﬁgure.

The frequency broadening in Brune et al ’s experiment

can be attributed to the multi-mode J-C model, ˆ

2~ω0σ3+P~

ks ~ω~

kˆa†

ksˆa~

ks −iP~

ks ~g~

ks[σ+ˆa~

ks −σ−ˆa†

ks][31]

[18,19], rather than the single-mode Jaynes-Cummings

model [20]. Thus the theoretical explanation of the CR

of the quantum Rabi oscillations in a lossy resonant cav-

ity needs a novel approach with the multi-mode Jaynes-

Cummings model. The novel approach must take losses

from the cavity into account for the explanation of Brune

et al ’s experimental data [7]. Here we provide a novel

theory for the CR by considering losses from the cav-

ity especially the “Ohmic” loss [21] to the walls of the

cavity, the leakage from the cavity, and the loss due to

the spontaneous emission through the open surface of the

cavity.

Multi-mode J-C model [18] is well-known [20,22,23] as

an extension of the single-mode J-C model [12]. Multi-

mode J-C model has been successfully used by us [20]

to explore the quantum Rabi oscillations of a two-level

system interacting with a very low average number of

injected coherent photons (¯n= 0.4) in a lossy resonant

cavity as described in Brune et al ’s experiment [7]. Such

a very low average number of photons was treated per-

turbatively (up to the second order in ¯n21) in Ref.

[20]. However, Brune et al [7] obtained two more sets of

data for low average number of injected coherent photon

numbers ¯n= 0.85 ±0.04 and ¯n= 1.77 ±0.15 in the same

cavity showing the CR of the quantum Rabi oscillations

arXiv:2210.04039v1 [quant-ph] 8 Oct 2022

FIG. 1: Schematic diagram for a two-level system interacting

with injected coherent photons in a lossy resonant cavity.

of a two-level system (87Rb atom). A non-perturbative

method is needed for the theoretical explanation of these

two sets of data for the CR. Hence we extend our method

described in Ref. [20] for this purpose. The CR of the

quantum Rabi oscillations, of course, were not discussed

in Ref. [20].

Calculation in this article essentially begins with Eqn.

(8) of Ref. [20]. This equation is an outcome of the

multi-mode J-C model and it is nothing but the net tran-

sition probability (P2→1(t)) which describes the quantum

Rabi oscillations in time (t) domain for a two-level sys-

tem interacting with coherent photons in a lossy reso-

nant cavity. This transition probability is a function of

time and a number of parameters including the renormal-

ized coupling constant which can be determined by the

mode quality factor of the cavity and the average num-

ber of coherent photons incident on the two-level system.

We determine the transition probabilities for the average

numbers of coherent photons ¯n= 0.85 and ¯n= 1.77 and

the mode quality factor Q= 7 ×107after determining

the renormalized coupling constants within a graphical

method. We compare our theoretical results with the ex-

perimental data obtained by Brune et al [7] and the ex-

isting theoretical results obtained within the single-mode

J-C model [7,8,14]. We also estimate the collapse time

and the revival time for ¯n= 0.85 and 1.77. Finally, we

conclude.

II. COLLAPSE AND REVIVAL

Let us consider a two-level system (atom/molecule) in

a lossy resonant optical cavity of the resonance frequency

ω0and the mode quality factor Q. The two-level system

is interacting with the coherent photons which are in-

jected through a hole on the cavity axis. Let the average

number of coherent photons injected on the two-level sys-

tem be ¯n. We consider the quantum Rabi oscillations of

the two-level system in the processes of the spontaneous

emission, the stimulated emission and the stimulated ab-

sorption. The quantum Rabi oscillations need the two-

level system to strongly interact with the injected pho-

tons of the cavity ﬁeld. The high mode quality factor

of the cavity ensures strong light-matter coupling. The

photon emitted from the two-level has a long life-time

(∼200 µs) in such a situation. The emitted photon re-

peatedly reﬂects back and forth with the mirrors of the

cavity before it leaks out through the holes on the axis of

the cavity or becomes absorbed (or scattered) in the walls

of the cavity resulting in the “Ohmic” loss [21]. However,

the curved surface of the cylindrical geometry of the cav-

ity is also kept open. This causes additional loss from

the cavity. This loss is associated with the spontaneous

emission from the two-level system through the curved

surface of the cavity [20]. The probability that an emit-

ted photon escapes from the cavity through the curved

surface is p0=2πrh

2πrh+2πr2=1

1+ r

hwhere ris the radius

of each of the mirrors of the cavity and his the sepa-

ration of the two mirrors [20]. All these losses result in

the net quality factor as Q0=1

Q+p0A(0)

ω0

[20] where A(0)

is the frequency broadening due to the natural decay in

the free space inside the cavity and ω0is the Bohr fre-

quency of the two-level system. Here A(0) is nothing but

the enhanced value of the Einstein Acoeﬃcient due to

the Purcell eﬀect [24]. Derivation of the net quality fac-

tor has been shown in Appendix A. Let us consider that

initially (t= 0) the two-level system was in the excited

state. Thus we get the net transition probability of two-

level system from the excited state (|ψ2i) to the ground

state (|ψ1i) at time t, as [20]

P2→1(t) = A(0) ∞

n=0

¯nn

n!e−¯n[n+ 1] ×

Z∞

ωn

(ω0/Q0)2

4[Ω2

n−ω2

n]+(ω0/Q0)2

sin2(Ωnt/2)

ΩnpΩ2

n−ω2

dΩn

(1)

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Multi-modeJaynes-CummingsmodelresultsforthecollapseandtherevivalofthequantumRabioscillationsinalossyresonantcavityNajirulIslamandShyamalBiswasSchoolofPhysics,UniversityofHyderabad,C.R.RaoRoad,Gachibowli,Hyderabad-500046,India(Dated:February7,2022)Wehavenumericallyobtainedtheoreticalresultsforthecol...

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Multi-mode Jaynes-Cummings model results for the collapse and the revival of the quantum Rabi oscillations in a lossy resonant cavity Najirul Islam and Shyamal Biswas

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