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Suppression of γ-photon absorption viaquantum interferencePetr Anisimov a , Farit Vagizov a b , Yuri Rostovtsev a , RustemShakhmuratov c & Olga Kocharovskaya a da Department of Physics and Institute for Quantum Studies , TexasA&M University , College Station, Texas 77843-4242, USAb Kazan State University , Kazan 420008, Russiac Kazan Physical Technical Institute, RAS , Kazan 420029, Russiad Institute of Applied Physics, RAS , Nizhny Novgorod 603120,RussiaPublished online: 01 Dec 2010.

To cite this article: Petr Anisimov , Farit Vagizov , Yuri Rostovtsev , Rustem Shakhmuratov & OlgaKocharovskaya (2007) Suppression of γ-photon absorption via quantum interference, Journal ofModern Optics, 54:16-17, 2595-2605, DOI: 10.1080/09500340701553048

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Suppression of c-photon absorption via quantum interference

PETR ANISIMOV*y, FARIT VAGIZOVyz, YURI ROSTOVTSEVy,RUSTEM SHAKHMURATOV and OLGA KOCHAROVSKAYAy

yDepartment of Physics and Institute for Quantum Studies,Texas A&M University, College Station, Texas 77843-4242, USA

zKazan State University, Kazan 420008, RussiaKazan Physical Technical Institute, RAS, Kazan 420029, Russia

Institute of Applied Physics, RAS, Nizhny Novgorod 603120, Russia

(Received 19 March 2007; in final form 3 July 2007)

We show that the interference effects (similar to electromagnetically inducedtransparency, which was widely studied earlier in electronic transitions in optics)may appear in g-radiation at nuclear transitions under the condition of nuclearlevel anticrossing. We demonstrate it also experimentally in optically thin samplesof FeCO3.

1. Introduction

The quantum interference effects (such as electromagnetically induced transparency(EIT), slow light, lasing without inversion, etc.) represent one of the most rapidlydeveloping fields of research in quantum optics. These effects appear in three-levelquantum systems driven by a strong coherent field at one transition and probed by aweak field at the adjacent transition. In the particular case of two quantum stateshaving the same energy, the transition between these degenerate levels can beefficiently driven by a dc-field. This situation can be realized, for example, underthe condition of level anticrossing (i.e. level mixing) in an external magnetic field.Quantum interference effects were widely studied both theoretically and experi-mentally in the optical range of frequencies at the electronic transitions mainlyin gaseous media [1] and more recently in solids [2].

A few years ago a level mixing induced transparency (LMIT) was reported for thefirst time at the nuclear g-ray transition [3]. A more detailed experimental andtheoretical analysis of this effect was provided in the following papers [4–7].The experiments were done in a FeCO3 crystal with an optical thickness for theresonant g-quanta of the order of 10. About 25% suppression of the resonant g-rayabsorption in 57Fe was observed under the condition of nuclear level crossing.This effect was attributed to the multiple scattering with a polarization change in theoptically thick sample [6, 7]. Hence it should not appear in thin samples.

*Corresponding author. Email: petr@tamu.edu

Journal of Modern OpticsVol. 54, Nos. 16–17, 10–20 November 2007, 2595–2605

Journal of Modern OpticsISSN 0950–0340 print/ISSN 1362–3044 online # 2007 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/09500340701553048

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In this paper we report an experimental demonstration of the level mixinginduced transparency for g-photons in a thin sample of FeCO3 and show that thiseffect can be viewed as EIT for single g-photons at the nuclear transitions.

2. The characteristics of the samples and the experimental results

We perform the Mossbauer experiments in the natural mineral siderite, FeCO3.The nuclear spin of 57Fe is equal to Ig ¼ 1=2 in the ground state and to Ie ¼ 3=2 inthe excited (14.4 keV) state. The siderite does not have cubic symmetry, thus it has anelectric field gradient which interacts with the nucleus’ quadrupole moment and splitsthe excited state by 23.84MHz. At the temperature below the Neel temperature[TN ¼ 38:3K] (see [8]) siderite becomes anti-ferromagnetic with an internal magneticfield, B0 being parallel or anti-parallel to the electric field gradient. This directionis naturally chosen as a quantization z axis. Thus, the Hamiltonian of the systemhas the form:

H ¼He Vg

Vg y

Hg

, ð1Þ

where He is a 4 4 Hamiltonian for an excited state manifold; Hg is a 2 2Hamiltonian for a ground state manifold, and Vg ¼ lgBg describes a magneticdipole interaction with a probing g-radiation, where lg is a magnetic dipole momentof the transition, and Bg is a magnetic field of a probing g-radiation.

The Hamiltonian for an excited state manifold includes magnetic and quadrupolecontributions He ¼ HZ þHQ which are in turn

HZ ¼ ge Ieð ÞzB0 ð2Þ

and

HQ ¼ Vq 3 Ieð Þ2zIe Ie þ 1ð Þ

, ð3Þ

where ge is the gyromagnetic ratio for an excited state. The Hamiltonian for a groundstate manifold includes only the magnetic contribution Hg ¼ g Ig

zB0, where gg is

the gyromagnetic ratio for a ground state. Vq ¼ eQVzz= 4I 2I 1ð Þð Þ, where Q isthe nuclear quadrupole moment and Vzz is zz component of the electrostaticgradient tensor.

The structure of the energy levels is given in figure 1. In the crystal samplethe transition probability depends on the geometry of the experiment. If a probeg-radiation propagates along the z axis, the transitions with m ¼ þ1(corresponding to the lines 1 and 5 in figure 1 and described by Vþ

g ¼ þððBgÞx þ

iðBgÞyÞ) and with m ¼ 1 (corresponding to the lines 3 and 4 and describedby V

g ¼ ððBgÞx iðBgÞyÞ) are allowed. In the case when a probe g-radiationpropagates in the direction perpendicular to the z axis, the transitions with m ¼ 0(lines 2 and 6; V0

g ¼ 0ðBgÞz) also become possible. Because of equal populations in

yHermitian conjugated.

2596 P. Anisimov et al.

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the ground state and a fundamental chiral symmetry of the electromagnetic inter-action, the transitions m ! m0 should be identical to the transitions m ! m0

(corresponding to the reverse direction of the z axis). Thus, for a single crystal sampledepending on the geometry of the experiment, the Mossbauer absorption spectrashould consist of either two (1–4 and 3–5) or three doublets (1–4, 3–5 and 2–6) withidentical lines (i.e. the same shape, width and amplitudes) in each doublet.

At T¼ 30K, when the internal magnetic field B0 is approximately 15T, Zeemansublevels 3=2 and 1=2 should cross each other and therefore the two lines 4 and 5should merge into one line. Thus, in the optically thin sample an absorption of themerging line should be equal to the sum of the absorption of lines 4 and 5, whichin turn is equal to the sum of 1 and 3 absorption lines.

On the other hand, in an optically thick sample one could expect that an increaseof the optical thickness of the merging line to lead to summation of the absorptioncoefficients in the exponent rather than to summation of two exponents with originalabsorption coefficients. It would result in absorption saturation of lines 4 and 5 attheir crossing point as compared to the sum of absorption of their partner’s lines 1and 3 at the exit of the absorbing medium. However, such a simple ‘thickness’argument does not hold in the case of a parallel geometry.

Indeed, the radioactive source in both our and the previous experiments wasunpolarized and thus could be thought of as emitting g-photons of right and leftcircular polarizations one by one following each other with an equal probability.In a parallel geometry each of these photons may excite only one correspondingtransition: either m ¼ þ1 (line 5) or m ¼ 1 (line 4) depending on the polariza-tion of the coming photon. Thus, the total absorption of the merging lines should bedetermined by the sum of 1 and 3 absorption lines at the exit of the sampleindependent of its thickness. However, the earlier experiments in the opticallythick samples with an optical thickness for the resonant g-quanta of an order of10 done at the Catholic Leuven University had shown that an absorption at the levelcrossing was about 25% smaller than the sum of absorptions at 1 and 3 linesindependent of the geometry.

B =15 T

I=3/2

I=1/2

HQ

Fe in FeCO57

3/2

1/2 –3/2–1/2

–1/2 1/2

11

2

33

6

55

4

0

3

Figure 1. Level diagram and possible transitions in siderite. The energy levels that wefocused on are shaded.

Suppression of g-photon absorption via quantum interference 2597

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It is important to note that we repeated the same experiment at Texas A&MUniversity with a crystal sample with an optical thickness of 7 for the resonantg-quantay in a parallel geometry and received a similar result (see figure 2). Such adeficit in absorption at the level crossing point was interpreted via the scattering ofg-photons with a polarization change in the optically thick crystal sample with abroken axial symmetry [6, 7]. If this scattering process and the simple thickness effectwas the only mechanisms for the observed transparency, the transparency shouldvanish in the optically thin sample.

We did the further Mossbauer experiments with the poly-crystal powder fortwo reasons. First, it is technically easier to prepare optically thin powder samples.Second, the scattering mechanism discussed in [6] should be eliminated in a powdersample where the orientation of the optical axis is random and hence all threedoublets are present in the spectrum. The experiments were performed by usinga conventional Mossbauer setup. It includes a source of g-radiation (Co57 inrhodium), an absorber (FeCO3 in powder form), and a detector. The results fordifferent optical thicknesses of an absorber are presented in figure 3. It clearly showsthat transparency is reduced with a decrease of the sample thickness. At the sametime an appreciable deficit in absorption (about 8%) appears even in the sample withan optical thickness for the resonant g-quanta equal to 0.8.

In the next section we provide a theoretical analysis of the level crossingtransparency effect in thin samples. We show that the mechanism of LMIT in thinsamples is essentially the same as EIT which was widely studied at the electronicoptical transitions.

3. Mechanism of LMIT in thin samples

In order to understand the origin of the absorption deficit at the level crossingcondition we consider a simplified three-level system involving only sublevels ofinterest. Namely, these are m ¼ 1=2 and m ¼ 3=2 sublevels, which are crossing inthe excited state, and a single m ¼ 1=2 sublevel in the ground state. Let us denotethem a, c and b accordingly (figure 4). Similar to previous papers [3–7] we supposethat the axial symmetry is broken. (It may be caused by the presence of eitherperpendicular components of the internal magnetic field or asymmetry of theelectric field gradient.) Breaking of the axial symmetry means coupling between thetwo degenerated upper levels a and c. Such a coupling results in the nuclearresponses to probe g-radiation simultaneously at cb and ab transitions. Theirinterference under certain conditions may lead to the reduced absorption.

yThe reason for the high transparency is the following. The recoilless fraction of radiationboth in the emitter and absorber is only about 70% while our detector registers both resonantand off-resonant gamma-quanta. Besides, emitted radiation is not polarized, while absorptionof each resonant line is specified for radiation with a particular polarization, left or rightcircularly polarized.

2598 P. Anisimov et al.

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–4

–20

24

6707580859095100

sing

le c

ryst

alpa

ralle

l geo

met

ry

FeC

O3

HH

F~1

5 T

Transmission, %

Vel

ocity

, mm

/s–1

.0–0

.50.

00.

51.

0707580859095100

sing

le c

ryst

alpa

ralle

l geo

met

ry

FeC

O3

HH

F~1

5 T

Transmission, %

Vel

ocity

, mm

/s

Figure

2.

Transm

ission

spectrum

ofasinglecrystalsample(left).TheFeC

O3singlecrystalsampleiscleaved

andpolished

onboth

sides;itisa

60–70mm

thickplateletof0:8

0:7cm

2area.Theangle

ofthecaxiswiththecleavageplanef10 11gisaboutp=

4,andtheg-raywavevectoris

alongthecaxis.Thisplateletispasted

uponathin

1.1mm

copper

plate

withg-rayaperture

of6mm

indiameter.Thecopper

plate

isscrewed

onto

acold

finger

oftheHeclosed-cyclecryostatfrom

CryoIndustries.Thesample’sopticalthicknessfortheresonantg-quanta

isabout7.Zoom

inatthelevel

crossingisshownontheright.

Suppression of g-photon absorption via quantum interference 2599

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–4 –2 0 2 4 6

70

75

80

85

90

95

100

te ~ 6.8

Tra

nsm

issi

on, %

Velocity, mm/s

~22% deficit

–4 –2 0 2 4 694

95

96

97

98

99

100

~10% deficitte=1.2

Tra

nsm

issi

on, %

Velocity, mm/s

–4 –2 0 2 4 6

90

92

94

96

98

100

~14% deficitte =2.1

6

54

3

2

1

Tra

nsm

issi

on, %

Velocity, mm/s

–4 –2 0 2 4 6

96

97

98

99

100

~8% deficitte =0.8

Tra

nsm

issi

on, %

Velocity, mm/s

Figure 3. Transmission spectra of 57Fe:FeCO3 powder samples for different optical thick-nesses (te) of an absorber. Dots are the observed absorption while solid lines are the fittingcurves based on the assumption that absorption at a merging line is equal to the sum ofindividual absorptions. Samples were cooled down to 30K.

Detuning (δ γ/ c)

Abs

orpt

ion

(arb

. uni

ts)

prob

e γa γ c

Ω

b

ca

–1.5 –1 –0.5 0 0.5 1 1.5 2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 4. Absorption curves for V0 probing g-radiation. (i) Dotted line corresponds to

O ¼ 0, c ¼ 1 and a ¼ 0:31; (ii) solid line corresponds to O ¼ 0:1644c, c ¼ 1 and a ¼ 0:31;(iii) dashed line corresponds to O ¼ 0:1644c, c ¼ 1 and a ¼ 1.

2600 P. Anisimov et al.

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Indeed, when the axial symmetry is broken, the effective Hamiltonian of the systemtakes the form (h¼ 1):

Heff ¼

a b cabc

!a Vþ OVþ 0 V

O V !a

24

35, ð4Þ

where O is a coupling due to the broken axial symmetry. Assuming that g-radiationinteracts with both m ¼ 1 and m ¼ þ1 transitions and taking into accountthat it is weak and does not perturb the populations (i.e. bb ¼ 1, aa ¼ cc ¼ 0),we obtain in the rotation wave approximation the following equations for theoff-diagonal matrix elements at the probed transitions:

@ab@t

¼ a þ ið Þab i Vþ0 þ Ocb

, ð5Þ

@cb@t

¼ c þ ið Þcb i V0 þ Oab

, ð6Þ

where ga and gc are decay constants for the off-diagonal density matrix elements,and is the detuning from the transition frequency [9].

To get a deeper insight into the physics of LMIT we consider first the casewhen the probe field interacts only with the m ¼ 1 transition (i.e. Vþ

0 ¼ 0).Then we come to the so-called scheme. The steady-state solution of the aboveequations takes the form:

ab ¼ O

a þ ið Þ

V0

c þ iþ Oj j2= a þ ið Þ ð7Þ

and

cb ¼ iV

0

c þ iþ Oj j2= a þ ið Þ: ð8Þ

Hence the susceptibility of the system is defined as:

ij j2

c þ iþ Oj j2= a þ ið Þð9Þ

or

ij j2

2

1þ A

Bþ þ iþ

1 A

B þ i

, ð10Þ

where

A ¼c a

c að Þ24O2

1=2 ð11Þ

and

B ¼a þ c

2 i O2

c a2

2 1=2: ð12Þ

Suppression of g-photon absorption via quantum interference 2601

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If the linewidths of both ab and cb transitions were equal to each other (a ¼ c),then for arbitrary coupling between the excited states the total line shape(equation (10)) presents the sum of two Lorentzians with centres separated by acoupling constant 2O and identical linewidths and amplitudes. This result has atransparent interpretation in the basis of the states dressed by O. Namely, it meansthat the total line results from the summation of the dressed level lines separated by2O at the crossing point (level anti-crossing), and interference effects (associated witha contribution of the coherence between the dressed states) do not play any role.This effect is also well known as the Autler–Townes effect (i.e. probing byg-radiation of the Stark splitting produced by O). If the linewidths at ab andcb transitions in our Mossbauer experiments were defined by the radiative decay,they would be equal to each other. Hence, the absorption deficit (if any) in this casecould be caused only by the level splitting and nuclear interference would not appear.

However, as it is clearly seen from the experimental data (see figure 2), thelinewidths for these transitions are different, namely, !3=2 ¼ 0:592mm s1 and!1=2 ¼ 0:260mm s1. In this case the total line shape (equation (10)) cannot beviewed as a sum of two Lorentzians produced by the dressed states, and theinterference effects play a crucial role. Let us consider the case when c > a(which corresponds to our experimental scheme). According to equations (10)–(12),in the weak coupling case, namely O c að Þ=2, the total linewidth representsitself as the difference of two Lorentzians both centred at the merging lines’frequency. The positive Lorentzian has a broad linewidth defined asc O2= c að Þ (which is reduced with an increase of O) and a negativeLorentzian has a narrower linewidth a þ O2= c að Þ which is power broadened.The absorption coefficient in the crossing point is proportional to c þ O2=a

1.

Thus, an absorption deficit due to the coupling is proportional to O2= cað Þ.Note that the deficit in absorption for a given gc in the case a < c is larger thanin the case a ¼ c. This means that an interference in absorption is destructive(coherence between the dressed states reduces the absorption).

According to our experimental data, the linewidth at the cb transition is abouttwo times broader than the linewidth at the ab transition (see figure 2). (It isworthwhile to point out that this fact was noticed earlier and connected to thebroadening of 1=2 ! 3=2 line by coupling of the nuclear spin to the fast fluctuatingelectronic spin [8, 10].) This experimentally measured linewidth results from aconvolution of both the radiative source and absorber lines. Taking into accountthat the linewidth of our source is known to be equal to 0.11mms1 we concludethat c=a 3:217.

The absorption curves for V0 probing g-radiation corresponding to the cases

(i) O ¼ 0 and (ii) O ¼ 0:1644c for c=a ¼ 3:217 are shown in figure 4. As it can beseen from the figure the absorption deficit is about 8%. The case (iii) O ¼ 0:1644cand a ¼ c is also presented for comparison. Note again that the deficitin absorption is larger in the case of c=a ¼ 3:217 than it would be in thecase a ¼ c, indicating a constructive contribution of the destructive interferenceinto the deficit of absorption.

In principle, in the case of stronger coupling, O c að Þ=2, the totalabsorption line could be viewed as the sum of two lines separated by 2O (similar

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to the case a ¼ c) though with the deformed line shapes (as compared with theLorentzian ones) due to interference effects.

In this case the line splitting should actually be noticeable (see figure 5).Moreover, such strong coupling would apparently appear in the overallMossbauer spectra and not just for merging lines (where it was observed experi-mentally).

Let us consider now the case when the linewidth at the transition probed byg-radiation is narrower than at the adjacent transition. It corresponds to ourexperimental scheme with a < c probed at m ¼ 1, i.e. ab, transition only.In this case the same formulas (see equation (10)) can be used with a replacementof gc by ga and ga by gc. So that in the weak coupling limit the positive Lorentzianis getting to be narrower than the negative one. Note that the deficit in absorptionfor a given ga caused by O is smaller for c > a in this case than for a ¼ cindicating a constructive contribution of the coherence between the dressed statesto the absorption. The absorption curves for Vþ

0 probing g-radiation correspondingto the same cases as were considered before for V

0 probe are given in figure 6.Note that an absorption deficit for Vþ

0 probe at O ¼ 0:1644 is smaller than for theV

0 probe.As has already been noticed above, in the case of the unpolarized Mossbauer

source and single crystal absorber in the parallel geometry, the total absorptionspectrum of a thin sample may be obtained just by summation of absorption for V

0

and Vþ0 probes (see figure 7). The absorption deficit for O ¼ 0:1644c in this case is

equal to 8%.Thus, we may conclude that nuclear interference effects should result in the

absorption deficit under the level anti-crossing condition.As has been mentioned above, a breaking of an axial symmetry may occur either

via electric quadrupole or magnetic interaction. However, a transverse component ofa magnetic field does not provide a direct coupling between the crossing sublevels

Detuning (δ γ/ c)

Abs

orpt

ion

(arb

. uni

ts)

–1.5 –1 –0.5 0 0.5 1 1.5 2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 5. Absorption curves for V0 probing g-radiation in the case of a strong coupling.

(i) Dotted line corresponds to O ¼ 0, c ¼ 1 and a ¼ 0:31; (ii) solid line corresponds toO ¼ c að Þ=2 ¼ 0:345, c ¼ 1 and a ¼ 0:31; (iii) dashed line corresponds to O ¼ 0:345,c ¼ 1 and a ¼ 1.

Suppression of g-photon absorption via quantum interference 2603

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(m ¼ 1=2 and m ¼ 3=2). It couples them only in the second order via the sublevelm ¼ 1=2 of the excited state manifold. Besides it leads to an additional couplingbetween the sublevels m ¼ 1=2 and m ¼ 3=2 in the excited state manifold and thesublevels m ¼ 1=2 and m ¼ 1=2 in the ground state manifold. As a result of this,along with an absorption deficit at the crossing point, it would lead simultaneously tothe appearance of additional lines in the Mossbauer spectra which were not observedexperimentally.

Thus, we focus on the case when mixing of the upper levels occurs via quadrupoleinteraction, which couples the crossing sublevels directly and does not lead to anyadditional couplings. In the case of quadrupole coupling O ¼ 31=2Vq, where ¼ ðVxx VyyÞ=Vzz. Hence, O ¼ 0:1644c corresponds to ¼ 0.067.

Detuning (δ γ/ )

Abs

orpt

ion

(arb

. uni

ts)

probe

γ a γ c

Ω

b

ca

- 1.5 - 1 - 0.5 0 0.5 1 1.5 2

0.5

1.0

1.5

2.0

2.5

3.0

Figure 6. Absorption curves for Vþ0 probing g-radiation. (i) Dotted line corresponds to

O ¼ 0, c ¼ 1 and a ¼ 0:31; (ii) solid line corresponds to O ¼ 0:1644c, c ¼ 1 and a ¼ 0:31;(iii) dashed line corresponds to O ¼ 0:1644, c ¼ 0:31 and a ¼ 0:31.

Detuning (δ γ/ c)

Abs

orpt

ion

(arb

. uni

ts) +×609.0 ×2

–1.5 –1 –0.5 0 0.5 1 1.5 2

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure 7. Net absorption curves for unpolarized g-radiation. (i) Dashed line corresponds toO ¼ 0, c ¼ 1 and a ¼ 0:31; (ii) solid line corresponds to O ¼ 0:1644c, c ¼ 1 and a ¼ 0:31.

2604 P. Anisimov et al.

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4. Conclusion

We experimentally observed a reduction of the resonant g-ray absorption

under the nuclear level crossing condition in the optically thin sample of FeCO3.

We analytically analysed a simplified three-level model (including two crossing upper

levels and the single lower state) assuming coupling between the crossing levels due

to the axial symmetry breaking in the system and proved that the nuclear interference

(similar to EIT at atomic transitions in optics) caused by this coupling may lead to

suppression of the Mossbauer absorption at the level crossing point. Finally, we

estimated that a weak breaking of an axial symmetry via quadrupole interaction,

¼ 0.067, would result in the observed 8% deficit of the resonant absorption.

Acknowledgement

We are deeply grateful to Romain Coussement, Jos Odeurs, and Stijn Gheysen for

many fruitful discussions of LMIT. We appreciate a support of this work by NSF

and AFOSR. We warmly thank Clayton Holle for his help in the preparation of the

manuscript.

References

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