Distribution of squeezed spin quantum noise in an antiferromagnet

ARTICLE IN PRESS

Physica B 348 (2004) 306–309

Distribution of squeezed spin quantum noise in an antiferromagnet Feng Penga,b,*, Bai Haob a

China Center of Advanced Science and Technology (World Laboratory), P.O. Box 8730, Beijing 100080, China b Department of Physics, Beijing University of Science and Technology, Beijing 100083, China Received 3 July 2003; received in revised form 15 December 2003; accepted 16 December 2003

Abstract We study the magnon-squeezed states generated by second-Raman scattering. The magnon-squeezed states allow a reduction in the spin-component fluctuation to below the level of the vacuum fluctuation. As the laser fields change the symmetry of the spin system, this leads to the anisotropy of the spin-fluctuation distribution. We evaluate the fluctuation of the spin component and discuss the anomalies of the spin-fluctuation distribution. r 2003 Elsevier B.V. All rights reserved. PACS: 75.10; 75.30.D; 76.30.D Keywords: Magnon-squeezed states; Spin fluctuation; Raman scattering; Quantum noise

The artificial modulation of the quantum noise has attracted much attention during the past few years. It is reported that the phonons and the magnons can realize the squeezed states under the external field manipulating, and the fluctuations of their corresponding vector fields are lower than the levels of their vacuum fluctuations [1–9]. However, much more efforts are still required to detect the squeezed quantum noise. The experimental results in Ref. [4] have not been reproduced by any other group. Here we mainly study the artificial manipulation of the spin-component fluctuation. *Corresponding author. Physics Department, Beijing University of Science and Technology, Beijing 100083, China. Tel.: +86-1062333425; fax: +86-1062327283. E-mail address: [email protected] (Feng Peng).

As the laser fields change the symmetry of the spin system in the second-order Raman scattering (SORS) experiment, this would lead to the anomalous spin-fluctuation distribution in k% space. In this work, we will study the fluctuation distribution in wave vector space by using the quantum theory. An effective dipole electric approximation is adopted to describe the interaction between the laser field and the spin waves in SORS [10,11]. In the antiferromagnet, the magnons can generate the self-squeezing states due to the sub-spin waves from two sublattices interfering each other. The laser fields in SORS take a role of modulating the squeezing states. By adjusting laser field strength, frequency and incidental direction, we can change the fluctuation distribution of spins in the squeezed states.

0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.12.014

ARTICLE IN PRESS Feng Peng, B. Hao / Physica B 348 (2004) 306–309

In the next section, we will deduce the formula of spin-x-component fluctuation and discuss the anomalies of the fluctuation distribution. In the antiferromagnet, the Heisenberg Hamiltonian is X , , X z H0 ¼ jJj Sai  Sbj gmB ðB þ BA Þ Sai i;j

 gmB ðB  BA Þ

X

i z Sbj ;

ð1Þ

i ,

,

where S ai and S bj are the spins at the ith and jth sites belonging the m and k sublattices, J the exchange integrals, mB the Bohr magneton, BA the , anisotropy field, and the external magnetic field B is parallel to the z-axis. The exchange interaction is limited to nearest neighbors of each other. Suppose the sublattice magnetization be along the z-axis. The Raman scattering interaction for light polarized in the xz-plane is [10,11] X HR ¼ C0 E 2 ðtÞ sinð2yÞ sgn i;j y y x x  ½ðxbj  xai Þðzbj  zai Þ ðSai Sbj þ Sai Sbj Þ;

ð2Þ

where EðtÞ ¼ E0 cosðotÞ is a continuous electric field, C0 is constant, and y is the angle between the electric field and the z-axis. The Hamiltonian of the system is H ¼ H0 þ HR : First of all, we introduce the raising and lowering operators: y y þ x  x Smi ¼ Smi þ iSmi ; Smi ¼ Smi  iSmi ; (m ¼ a; b), and then make use of the Holstein–Primakoff þ  transformation: Smi -ð2SÞ1=2 mi ; Smi -ð2SÞ1=2 mþ i þ z and Smi -S  mi mi ; (m ¼ a; b). Finally, we introduce a transformation to , spin wave , P variables , through ak ¼ N 1=2 i eik R i ai ; bk ¼ , P i k R i N 1=2 e b ; and their corresponding conjuj

j

gates. In this way, the Hamiltonian can be rewritten as X H¼ f_oa ak ak þ _ob bk bk k

þ ½Ak þ Bk ðe2iot þ e2iot Þ ðbk ak þ ak bk Þg; ð3Þ where _oa ¼ 2ZSjJj þ gmB ðBA þ BÞ; 2ZSjJj þ gmB ðBA  BÞ;

_ob ¼

Ak ¼ 16SZjJj cosðkx aÞ cosðky bÞ cosðkz cÞ  4C0 E02 sinð2yÞ sinðkx aÞ cosðky bÞ sinðkz cÞ;

307

Bk ¼ 2C0 SE 2 sinð2yÞ sinðkx aÞ cosðky bÞ sinðkz cÞ; and a; b; c are basic vectors in the lattices. The time evolution of the state vector jtSk with the wave vector k~ obeys jtSk ¼ Uk ðtÞj0Sk ; where j0Sk is the initial state (vacuum state). The timeevolution operator Uk ðtÞ can be expressed as þ þ þ þ  ð4Þ Uk ¼ eiðoa ak ak þob bk bk Þt eBk ak bk Bk ak bk ; where Ak ½eiðoa þob Þt  1

_ðoa þ ob Þ Bk ½eiðoa þob þ2oÞt  1

þ _ðoa þ ob þ 2oÞ Bk ½eiðoa þob 2oÞt  1

þ ð5Þ _ðoa þ ob  2oÞ and its conjugate is xk : Both xk and xk are squeezed factors. We find that the squeezed factors are not related to the external fields only on the planes kx ¼ 0; 7p=a and kz ¼ 0; 7p=c in k~ space. In other words, the external fields do not generate any modulation to the magnon-squeezed states on these planes. We can obtain the time-evolution operators (Heisenberg operators) qffiffiffiffiffiffiffiffiffiffi a# k ðtÞ ¼ Uk1 ak Uk ¼ eioa t coshð xk xk Þa# k sffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi xk ioa t sinhð xk xk Þb#k ;  e ð6Þ xk xk ¼

qffiffiffiffiffiffiffiffiffiffi b#k ðtÞ ¼ Uk1 bk Uk ¼ eiob t coshð xk xk Þb#k sffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi xk iob t sinhð xk xk Þa# k :  e xk

ð7Þ

The fluctuation of the spin x-axis component at time t is expressed as /DSx2 S ¼ /Sx2 S  /Sx S2 S X X ik~R~m ¼ e /tjak ak þ ak ak þ bk bk 2 k m þ b b þ 2ða b þ a b ÞjtS k k

k k

k k

S X X ik~R~m e /0jak ðtÞak ðtÞ ¼ 2 k, m þ ak ðtÞak ðtÞ þ bk ðtÞbk ðtÞ þ bk ðtÞbk ðtÞ þ 2½a ðtÞb ðtÞ þ a ðtÞb ðtÞ j0S: ð8Þ k

k

k

k

ARTICLE IN PRESS Feng Peng, B. Hao / Physica B 348 (2004) 306–309

308

We introduce Gk ðtÞ as the spin-component fluctuation in the wave vector space, namely G ðtÞ ¼ 1 /tja a þ a a þ b b 2

k k

k k

k k

þ bk bk þ 2ðak bk þ ak bk ÞjtS:

ð9Þ

Eqs. (6) and (7) are substituted into Eq. (9), we obtain qffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi Gk ðtÞ ¼ 1 þ 2 sinh2 ð xk xk Þ  sinhð xk xk Þ qffiffiffiffiffiffiffiffiffiffi "sffiffiffiffiffiffi xk iðoa þob Þt e  coshð xk xk Þ xk 3 sffiffiffiffiffiffi xk iðoa þob Þt 5 : ð10Þ þ e xk The condition of achieving the magnon-squeezed states is Gk ðtÞo1: Take the antiferromagnet material Mn F2 as an example. The parameters are chosen as jJj ¼ 11:9 mev; Z ¼ 6; S ¼ 52; BA ¼ 0:88 T and o ¼ 0:5ðoa þ ob Þ [12]. In Fig. 1, the solid line represents the fluctuation of spin x-axis component versus time under the laser field with the field strength E ¼ 1:0  102 V=cm at the point of k~ ¼ ð0:1p=a; 0:49p=b; 0:1p=cÞ; and the dot line to the fluctuation of spin x-axis component at the same wave vector in the absence of the laser field. It is very apparent that the laser fields effectively squeeze the spin quantum noise with more than 20% reduction. However in the identical laser fields, the laser fields do not generate any modulation to the spin fluctuation on the planes kx ¼ 0; 7p=a; and kz ¼ 0; 7p=c in k~ space. Although the laser fields may stimulate the squeezing of the spin-component fluctuation, this effect can take place only on the particular domains in the wave vector space. This shows that the laser fields have preferentially squeezing to the spin-component fluctuation with different wave vectors. This effect would affect the distribution of some measurable quantities, such as the spin susceptibility tensor. As we know, the diagonal component of the spin susceptibility tensor may be written as wxx ðk~; tÞ ¼ 2 2 x 2 g mB =V /ðSk Þ S; and it can also be expressed as g2 m2B S Gk ðtÞ: wxx ðk~; tÞ ¼ V

ð11Þ

Gk(t)

k

1.58

1.17

0.76 0

1 t(×10−14s)

2

Fig. 1. Fluctuations of spin x-axis component Gk ðtÞ versus time. The material parameters are chosen as jJj ¼ 11:9 mev; Z ¼ 6; S ¼ 52; BA ¼ 0:88 T; o ¼ 0:5ðoa þ ob Þ; C0 ¼ 1  1022 and the wave vector k~ ¼ ð0:1p=a; 0:49p=b; 0:1p=cÞ: The solid line corresponds to the external field strength E ¼ 1:0  102 V=cm; and the dot line to E ¼ 0:

For the laser fields modulate the spin-component fluctuation Gk ðtÞ only on some domains of the wave vector space, it can change the distribution of wxx ðk~; tÞ on the same domains. In conclusion, the second-order Raman scattering can efficiently affect the fluctuation of spins except some particular geometrical planes in the wave vector space. On the planes kx ¼ 0; 7p=a and kz ¼ 0; 7p=c in k~ space, the laser fields do not couple with the spins, and they do not generate any nodulation to the magnon-squeezed states on these planes. The laser field’s induced squeezing to the spin quantum noise would change the distribution of the spin susceptibility in wave vector space.

Acknowledgements This work was supported by National Natural Science Foundation of China.

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