Phonon Squeezed States: Quantum Noise Reduction in Solids

Physica B 263—264 (1999) 16—29

Phonon squeezed states: quantum noise reduction in solids

Xuedong Hu!, Franco Nori" *

!Department of Physics, University of Maryland, College Park, MD 20742, USA "Department of Physics, The University of Michigan, Ann Arbor, MI 48109-1120, USA

Abstract

This article discusses quantum fluctuation properties of a crystal lattice, and in particular, phonon squeezed states. Squeezed states of phonons allow a reduction in the quantum fluctuations of the atomic displacements to below the zero-point quantum noise level of coherent phonon states. Here we discuss our studies of both continuous-wave and impulsive second-order Raman scattering mechanisms. The later approach was used to experimentally suppress (by one part in a million) fluctuations in phonons. We calculate the expectation values and fluctuations of both the atomic displacement and the lattice amplitude operators, as well as the effects of the phonon squeezed states on macroscopically measurable quantities, such as changes in the dielectric constant. These results are compared with recent experiments. Further information, including preprints and animations, are available in http://www-personal.engin.umich.edu/ &nori/squeezed.html. ( 1999 Elsevier Science B.V. All rights reserved.

Keywords: Squeezed states; Quantum noise; Noise reduction; Phonon squeezed states

1. Introduction Phonons can also be excited phase-coherently. For instance, coherent acoustic waves with fre- Classical phonon optics [1,24—33] has succeeded quencies of up to 10 Hz can be generated by in producing many acoustic analogs of classical piezoelectric oscillators. Lasers have also been used optics, such as phonon mirrors, phonon lenses, to generate coherent acoustic and optical phonons phonon filters, and even ‘phonon microscopes’ that through stimulated Brillouin and Raman scattering can generate acoustic pictures with a resolution com- experiments. Furthermore, in recent years, it has parable to that of visible light microscopy. Most been possible to track the phases of coherent op- phonon optics experiments use heat pulses or tical phonons (for excellent reviews on coherent superconducting transducers to generate incoherent phonons, see, e.g., Refs. [2,34]) due to the availabil- phonons, which propagate ballistically in the crys- ity of femtosecond-pulse ultrafast lasers (with a tal. These ballistic incoherent phonons can then be pulse duration shorter than a phonon period), and manipulated by the above-mentioned devices, just techniques that can measure optical reflectivity like in geometric optics. with accuracy of one part in 10. In most situations involving phonons, a classical description is adequate. However, at low enough tem- * Corresponding author. peratures, quantum fluctuations become dominant.

For example, a recent study [3] shows that quan- in Refs. [10,12,41] and the impulsive case studied in tum fluctuations in the atomic positions can indeed Refs. [12,14]. influence observable quantities (e.g., the Raman Squeezed phonons could be detected by meas- line shape) even when temperatures are not very uring the intensity of the reflected or transmitted low. With these facts in mind, and prompted by the probe light [10—12,14,41]. This method has been many exciting developments in classical pho- used to detect coherent phonon amplitudes, since non optics, coherent phonon experiments, and (on reflectivity and transmission are closely related to the other hand) squeezed states of light (see, e.g., the the atomic displacements in a crystal. Measuring special issue on squeezed states in Ref. [4]), we a transmitted probe light pulse, Garett et al. [14] would like to explore phonon analogs of quantum tried to observe squeezed phonons produced by an optics. In particular, we study the dynamical and impulsive SORS. Stronger evidence might be avail- quantum fluctuation properties of the atomic dis- able in the future. The intensity of the CW SORS placements, in analogy with the modulation of signal for many materials might be too weak to be quantum noise in light. Specifically, we study single- detected with current techniques, but might be ac- mode and two-mode phonon coherent and squeezed cessible in the future. states, and then focus on a second-order-Raman- scattering-based approach to achieve smaller quantum noise than the zero-point fluctuations of the 2. Analogies and differences between phonons atomic lattice. and photons The concepts of coherent and squeezed states were both originally proposed in the context of quan- Coherent and squeezed states were initially in- tum optics. A coherent state is a phase-coherent troduced to describe photons. Here we are inter- sum of a number of states. In this state, the quan- ested in applying these concepts to phonons. tum fluctuations in any pair of conjugate variables Although both photons and phonons are bosons, are at the lower limit of the Heisenberg uncertainty they do have significant differences, and the physics principle. In other words, a coherent state is as of squeezed states of light cannot be straightfor- ‘quiet’ as the vacuum state. Squeezed states [4] are wardly extended to phonons. Table 1 is a chart interesting because they can have smaller quantum presenting a schematic comparison between pho- noise than the vacuum state in one of the conjugate nons and photons. Below, we briefly mention a few variables, thus having a promising future in differ- important similarities and differences that are rel- ent applications ranging from gravitational wave evant to our study. detection to optical communications. In addition, Photons are elementary particles with no inter- squeezed states form an exciting group of states and nal structure, thus are sometimes called simple can provide unique insight into quantum mechan- bosons. On the other hand, phonons describe the ical fluctuations. collective displacements of very many atoms in In recent years, squeezed states are also being a crystal, and are thus sometimes described as com- explored in a variety of non-quantum-optics sys- posite bosons [15]. Phonons are bosons because of tems, including ion-motion and classical squeezing the commutation relation between the coordinate [5,35,36] molecular vibrations [6,37,38], polaritons and momentum operators. Kohn and Sherring- [7,8,39], and phonons in crystals [9—13,40,41]. ton [15] pioneered the research on composite bo- Refs. [10—12,41] propose a second-order Raman sons like phonons, excitons, etc., and classified scattering (SORS) process for phonon squeezing: if them into two categories, with type-I referring to the two incident light beams are in coherent states, those bosons composed of an even number of fer- the phonons generated by the SORS are in a two- mions (such as He atoms), and type-II referring to mode squeezed state. those that are collective excitations — such as Here we first present an introduction to the sub- phonons, excitons, magnons, etc. In this sense it is ject of coherent and squeezed phonons, and later also possible to consider photons to be type-II on consider both the continuous wave case studied composite bosons [15], because they are the energy 18 X. Hu, F. Nori / Physica B 263—264 (1999) 16—29

Table 1 Comparison of several physical properties of phonons and photons

Phonon Photon

Type of boson type-II composite simple Propagating media discrete continuous Interactive ? yes yes in nonlinear media, not in linear media Massive ? yes no Macroscopic description wave eq. for elastic continuum Maxwell equations Microscopic description Schro¨ dinger eq. Quantum Electrodynamics Number of normal modes per allowed k 3p modes for each k 2 modes for each k u"u u" Dispersive ? always: Q(k) not in vacuum: ck Restriction on wave vector k confined to 1st Brillouin zone arbitrary Linear momentum ? vanishes non-zero Spin not defined s"1 quanta of electromagnetic field modes. Their The dispersion relations for photons and pho- commutation relation originates from the simple nons are qualitatively different. Photons in free harmonic oscillators that are used to quantize the space have a linear dispersion relation. On the electromagnetic field. Essentially, both photons other hand, phonons have complicated nonlinear and phonons are field quanta: photons are quanta dispersion relations which generally have several of a continuous field, while phonons are quanta of acoustic and optical branches. The acoustic a discrete field. branches are linear around the center of the first Non-interacting phonons are used to describe Brillouin zone, i.e., the k"0 point, which is at the harmonic crystal potentials. However, anharmoni- continuum limit. When the quasi-wave-vector k is city, which leads to phonon—phonon interactions, close to the first Brillouin zone boundary, u satu- is always present. Some properties of solids, such as rates. The optical branches of the phonons have lattice heat conductivity and thermal expansion, a different profile. Their dispersion relations are flat " u"u solely depend on the anharmonic terms in the crys- near k 0, where . Furthermore, as k in- tal potential. In other words, phonons in general creases, u decreases; indeed, the optical phonon interact with each other. For photons, the situation dispersion relation can be even more complicated is somewhat different. In vacuum and at low inten- depending on the lattice structure. Compared to sity, photon interactions are so weak that the rule photons, which generally have relatively simple dis- of linear superposition holds. However, in nonlin- persion relations, phonons have nonlinear disper- ear media, photons are effectively interactive, with sion relations that make it more difficult to satisfy their interaction mediated by the atoms. both energy and momentum (in fact, quasi-momen- As mentioned above, phonons exist in discrete tum) conservation laws simultaneously. media. Therefore, phonons have cut-off frequencies, The order of magnitude of the crystal cohesion which put an upper-limit to their energy spectra. energy determines that phonons have very low For a diatomic lattice, this limit is of the order of energies. In addition, phonons can easily couple 0.1 eV, which is in the infrared region. Photons, on to many other excitations which are in a similar the other hand, do not have such an upper bound energy range, and be perturbed by thermal fluctu- for their energy. In addition, the discrete atomic ations even at low temperatures. All these coup- lattice and the massive atoms lead to a finite zero- lings make phonon dynamics very dissipative. Due point fluctuation in the phonon field, while the con- to its strong damping, coherent phonons have very tinuous photon modes and the massless photons short lifetimes (&50 ps for optical phonons, while contribute to a divergent zero-point fluctuation in larger for acoustic phonons) [2,34]. On the other the photon field. hand, there exist many materials in which photons X. Hu, F. Nori / Physica B 263—264 (1999) 16—29 19 can propagate with little dissipation, and further- When no phonon is excited, the crystal lattice is more very long photon coherent times can be pro- in the phonon vacuum state "02. The expectation duced by lasers. values of the atomic displacement and the lattice To summarize this brief comparison, we notice amplitude are zero, but the fluctuations will be that the differences between phonons and photons finite: can often hinder our effort to apply ideas origin- 1 * 2 ,1 2 !1 2 ating in quantum optics to phonons. For example, ( uG?) !# (uG?) !# uG? !# (5) good phonon cavities at the moment does not exist, , "º " so it is very difficult to select phonon modes. For "+ q? , (6) Nm u phonons, we have to almost always deal with a q 2 q? continuum of phonon modes. In addition, short 1(*u($q))2 "2. (7) phonon lifetimes and the strongly dissipative envi- !# ronment of phonons also complicate the problems. The eigenstates of the harmonic phonon Hamil- " 2 These will be taken into consideration when we tonian are number states which satisfy bq nq " work on the theory. ( " 2 nq nq!1 . The phonon number and the phase of atomic vibrations are conjugate variables. Thus, due to the uncertainty principle, the phase is arbit- 3. Phonon operators and the phonon vacuum rary when the phonon number is certain, as it is the " 2 and number states case with any number state nq . Thus, in a number state, the expectation values of the atomic displace- A phonon with quasimomentum p" q and 1 " " 2 ment nq uG? nq and q-mode lattice amplitude j e " u 1 " $ " 2 branch subscript has energy qH qH; the corre- nq u( q) nq vanish due to the randomness in the sponding creation and annihilation operators sat- phase of the atomic displacements. The fluctuations " 2 isfy the boson commutation relations in a number state nq are [b , bR ]"d d ,[b , b ]"0. (1) "º " , "º " qYHY qH qqY HHY qH qYHY q nq q 1(*u )2 " ? # + Y? , (8) G? Nm u 2Nm u The atomic displacements uG? of a crystal lattice are q? qY$q qY? given by 1 * $ 2 " # ( u( q)) 2 2nq. (9) 1 , " + ºH H *q RG uG? q?Qqe . (2) (Nm qH 4. Phonon coherent states Here RG refers to the equilibrium lattice positions, a H to a particular direction, and Qq is the phonon A single-mode (q) phonon coherent state is an normal-mode operator eigenstate of a phonon annihilation operator: "b 2 b "b 2 bq q " q q . (10) QH" (b #bR ). (3) q 2u qH \qH qH It can also be generated by applying a phonon b For simplicity, hereafter we will drop the branch displacement operator Dq( q) to the phonon vac- j º uum state subscript , assume that q? is real, and define a q-mode dimensionless lattice amplitude operator: "b 2 b " 2 b R bH " 2 q "Dq( q) 0 "exp( qbq! q bq) 0 (11) R R u($q)"bq#b q#b q#bq. (4) \ \ "b " bLq "exp ! q + q "n 2. (12) This operator contains essential information on the A 2 B ( q Lq nq! lattice dynamics, including quantum fluctuations. It is the phonon analog of the electric field in the As is shown above, a phonon coherent state is a photon case. phase-coherent superposition of number states. 20 X. Hu, F. Nori / Physica B 263—264 (1999) 16—29

Moreover, coherent states are a set of minimum- These results can be straightforwardly generalized uncertainty states which are as noiseless as the to multi-mode coherent states. vacuum state. Coherent states are also the set of Coherent phonons have been the subject of con- quantum states that best describe the classical har- siderable interest in recent years. Typically, the monic oscillators [16]. dynamics of coherent phonons are described by A single-mode phonon coherent state can be using classical equations of motion. Here we pres- generated by the Hamiltonian ent a quantum description and show that it is consistent with the classical one and, as an additional " u R # #jH #j R H q(bqbq ) q (t)bq q(t)bq (13) bonus, contains information on quantum fluctuations. j and an appropriate initial state. Here q(t) repres- Coherent phonons can be generated by a femto- ents the interaction strength between phonons and second short pulse laser [10,41]. A femtosecond the external source. More specifically, if the initial pulse duration is much shorter than any phonon "t 2"" 2 state is a vacuum state, (0) 0 , then the state period and therefore acts as a delta-function driving vector becomes a single-mode coherent state there- force. It can produce coherent longitudinal optical after (LO) phonons [2,10,34,41]. We can make a very simplified calculation by replacing the coupling "t 2""K \*SqR2 (t) q(t)e , (14) j d ! strength q(t) with A (t t) in Eq. (13), so that where " u R # d ! H#) qbqbq A (t t)bq i R K (t)"! j (q)e*SqO dq (15) # Hd ! R q q A (t t)bq. (19) \ Here A""A"e*( is a time-independent complex am- is the coherent amplitude of mode q. If the initial plitude containing the information of the photon— state is a single-mode coherent state "t(0)2""a 2, q phonon interaction and the coherent amplitude of then the state vector at time t takes the form the relevant modes in the incident optical pulse. We "t(t)2""+K (t)#a ,e\*SqR2, (16) assume that the crystal is in the phonon vacuum q q " state before it is hit by the laser pulse at t t. º which is still a coherent state. The time-evolution operator (t, t) for the In a single-mode (q) coherent state phonon mode can be written as "K (t)e\*SqR2, 1u (t)2 and 1u($q)2 are q *? #) #) º " ! u R sinusoidal functions of time. The fluctuation in the (t, t) exp( i qtbqbq) atomic displacements is iA iAH ;exp ! b e\*SqR! bRe*SqR . (20) , "º " A 2 q 2 q B 1(*u )2 "+ q? . (17) G? #) 2Nmu q q? ' In other words, for t t, the crystal is in a coher- "K \*SqR2 The unexcited modes are in the vacuum state and ent state qe . The coherent phonon ampli- K thus all contribute to the noise in the form of zero tude q is point fluctuations. Furthermore, iAH K "! e*SqR, (21) 1 * $ 2 " q ( u( q)) #) 2. (18) 2 1 * 2 From the expressions of the noise ( uG?) #) and which is a constant complex number. Thus, a very 1 * $ 2 ( u[ q)] #), it is impossible to know which short laser pulse conveniently provides a time-inde- state (if any) has been excited, while this informa- pendent amplitude and a coherent phase to the ' tion is clearly present in the expression of the expec- active phonon mode(s). For t t, the average of 1 $ 2 tation value of the lattice amplitude u( q) #). the atomic displacement operator in the state X. Hu, F. Nori / Physica B 263—264 (1999) 16—29 21

"K \*SqR2" 2 qe 0\q becomes

1u 2" º K \*SqR *q RG G? u ( q? qe e 2Nm q

#º KH *SqR \*q RG \q? q e e )

2 "! "º K " u q? q Nm q

; u ! ! ) ! ! sin[( O(t t) q RG 3]. (22)

These longitudinal optical phonons can have a coherence time of about 50 ps at 10 K, and even longer at lower temperatures [2,34]. In the classical sense, ‘coherent’ means a wave with a well-defined phase, or waves that can inter- fere with each other when superimposed. Here we have shown that a single-mode coherent state of phonons generated by a short laser pulse is indeed a plane wave with a well-defined phase. Thus, these phonons in a quantum coherent state are also coher- Fig. 1. Schematic diagram of the uncertainty areas (shaded) in ! ! ent in a classical manner. Furthermore, they are in the generalized coordinate and momentum (X(q, q), P(q, q)) phase space of (a) the phonon vacuum state, (b) a phonon a minimum-uncertainty state. number state, (c) a phonon coherent state, and (d) a phonon squeezed state. Here X(q,!q) and P(q,!q) are the two-mode ($q) coordinate and momentum operators, which are the direct generalizations of their corresponding single-mode operators. 5. Phonon squeezed states Notice that the phonon coherent state has the same uncertainty area as the vacuum state, and that both areas are circular, while In order to reduce quantum noise to a level the squeezed state has an elliptical uncertainty area. Therefore, h below the zero-point fluctuation level, we need in the direction parallel to the /2 line, the squeezed state has a smaller noise than both the vacuum and coherent states. to consider phonon squeezed states. Quadrature squeezed states are generalized coherent states [14,42]. Here ‘quadrature’ refers to the dimensionless coordinate and momentum. Compared to co- a two-mode quadrature phonon squeezed state is herent states, squeezed ones can achieve smaller generated as variances for one of the quadratures during certain "a , a , m2"D (a )D (a )S (m)"02. (24) time intervals and are therefore helpful for decreas- q q q q q q q q a ing quantum noise. Figs. 1 and 2 schematically Here Dq( q) is the coherent state displacement oper- a "a " *( illustrate several types of phonon states, including ator with q" q e , vacuum, number, coherent, and squeezed states. mH m These figures are the phonon analogs of the illu- m R Sq( )"exp bq ! bq , (25) minating schematic diagrams used for photons A 2 2 B [17,42]. S (m)"exp(mHb b !mbR bR ) (26) A single-mode quadrature phonon squeezed q q q q q q state is generated from a vacuum state as are the single- and two-mode squeezing operator, and m"re*F is the complex squeezing factor with "a m2 a m " 2 h p q, "Dq( q)Sq( ) 0 ; (23) r*0 and 0) (2 . The squeezing operator 22 X. Hu, F. Nori / Physica B 263—264 (1999) 16—29

The time-evolution operator (to first order in m) for such a Hamiltonian has the form i º t " ! H t ( ) expA B ; mH !m R R exp[ (t)bqbq (t)bqbq], (28) where H " u bR b # u bR b , (29) q q q q q q i R m t " f q *(Sq>Sq)O q ( ) ( )e d . (30) \ Here m(t) is the squeezing factor and f(t) is the strength of the interaction between the phonon system and the external source; this interaction allows the generation and absorption of two phonons at a time. The two-mode phonon quadrature operators have the form ! " \ # R# # R X(q, q) 2 (bq bq b\q b\q) "2\u($q), (31) Fig. 2. Schematic diagram of the time evolution of the expecta- P(q,!q)"!i2\(b !bR#b !bR ). (32) tion value and the fluctuation of the lattice amplitude operator q q \q \q u($q) in different states. Dashed lines represent 1u($q)2, while We have considered two cases where squeezed 1 $ 2$ the solid lines represent the envelopes u( q) states were involved in modes $q. In the first case, (1[*u($q)]2. (a) The phonon vacuum state "02, where $ 1 $ 2" 1 * $ 2" the system is in a two-mode ( q) squeezed state u( q) 0 and [ u( q)] 2. (b) A phonon number "a a m2 m" *F " 2 1 $ 2" 1 * $ 2" q , q , ,( re ), and its fluctuation is state nq, n\q , where u( q) 0 and [ u( q)] \ # # "a 2 2(nq n\q) 2. (c) A single-mode phonon coherent state q , h h 1u $ 2" "a " u t a 1 * $ 2" \P # P where ( q) 2 q cos q (i.e., q is real), and [ u( q)] 2Ae cos e sin B. (33) 1(*u[$q)]2"2. (d) A single-mode phonon squeezed state 2 2 "a \*SqR m 2 m \*SqR qe , (t) , with the squeezing factor (t)"re and 1 2 "a " u 1 * 2 In the second case, the system is in a single-mode r"1. Here, u($q) "2 q cos qt, and [ u($q)] " \P u P u "a m2 a ""a " *( 2(e cos qt#e sin qt). (e) A single-mode phonon squeezed state q , ( q q e ) in the first mode "b 2 squeezed state, as in (d); now the expectation value of u is and an arbitrary coherent state q in the second 1 $ 2" "a " u a \ u( q) 2 q sin qt, (i.e. q is purely imaginary), and the mode. The fluctuation is now fluctuation 1[*u($q)]2 has the same time-dependence as in (d). Notice that the squeezing effect now appears at the times h 1[*u($q)]2"1#eP sin # when the lattice amplitude 1u($q)2 reaches its maxima, while A 2B in (d) the squeezing effect is present at the times when 1u($q)2 is close to zero. h #e\P cos # . (34) A 2B In both of these cases, 1[*u($q)]2 can be smaller than in coherent states (see Fig. 2). S (m) can be produced by the following Hamil- q q tonian: 6. Second-order Raman scattering (SORS) H " u bR b # u bR b q q q q q q q q So far we have focused on concepts and the #f R R #fH (t)bqbq (t)bqbq. (27) actual quantum mechanical states relevant to X. Hu, F. Nori / Physica B 263—264 (1999) 16—29 23 coherent and squeezed phonons. Now we will Hamiltonian has the form focus on one particular approach to generate (#)" !j + *SqR>*(# , squeezed phonon states via second-order Raman Hq Hq q bqb\qe c.c. , scattering. 1 j + q \q The SORS process originates from the quadratic q" P E E , (35) d 16 ?@ ? @ term in the polarizability change P?@ of a crystal. ?@ The photon—phonon interaction » that leads to the where and j refer to the overall phase and SORS process is [18] »"!+ +,+ q ?@ q HHY amplitude, respectively, of the product of the sec- PqH \qHYQ Q E E . Here, E and E are elec- ?@ qH \qHY ? @ ? @ ond-order polarizability and the incident electric tric field amplitudes along a and b directions with fields. Recall that Pq \q is real, therefore the phase frequencies u and u . The second-order polariza- ?@ has no q-dependence. It originates solely from bility tensor PqH \qHY satisfies PqH \qHY"P\qHY qH" ?@ ?@ ?@ the two photon modes. The Schro¨ dinger equation P\qH qHY. Recall that the complex phonon normal ?@ for the $q-mode phonons is i j "t (t)2" mode operator Q of the phonons is related to R q qH H(#)(t)"t (t)2, and its time-evolution operator can the phonon creation bR and annihilation b oper- q q \qH qH be solved by a transformation into the interaction ators by Q "b #bR . If the incident photon qH qH \qH picture. The result can be expressed as [10,12,41] fields are not attenuated we can treat the optical + , + H R R , fields as classical waves, and also consider the "t 2 &qR* Dq@q@\q\Dq@q@\q "t 2 q(t) "e e q(0) , (36) different pairs of $q modes as independent, f j \*( and treat them separately. Thus, for one par- where q"!i qte / . Notice that the second ticular pair of $q modes, the complete factor in the time-evolution operator is a two-mode Hamiltonian for the two phonon modes in- quadrature squeezing operator [21]. volved in the SORS process has the form [18]: In the CW case considered here, the amplitude " !+ \+ q \q , f Hq Hq 4 ?@P?@ E?E@ QqQ\q,whereof the squeezing factor q grows linearly with time. " u + R # R , Hq q bqbq b\qb\q is the free phonon However, this initial linear growth will be eventu- u Hamiltonian for the modes q and !q, q" ally curbed by subsequent phonon—phonon scat- u !u ( )/2, and the branch labels j and j have tering and optical pump depletion. In other words, f been dropped. the expression for the squeezing factor q is valid for Here we consider two different cases. The first is times much larger than one phonon period, but when the incident photons are in two monochro- much smaller than phonon lifetimes (because this matic beams [10,12,41]; i.e., with electric fields treatment considers non-decaying phonons). By " u # " EH EH cos( Ht H), j 1, 2. In the second case solving the phonon Langevin equation, we have the incident photons are in an ultrashort pulse shown that the squeezing factor in the CW SORS whose duration is much shorter than the phonon will eventually saturate at a constant value deter- period [12,14]. mined by the strength of SORS and the phonon decay constant [22]. In addition, the phase of the squeezing factor is determined by the phase difference of the two incoming light waves. If the $q 7. Squeezed phonons via continuous wave SORS phonon modes are initially in a vacuum state or in a coherent state, the SORS will drive them into a Let us now first consider the continuous wave two-mode quadrature squeezed state [10,12,41]. (CW) case. Because the photons are monochro- The time evolution operator of all the phonon matic, we can take a rotating wave approximation mode pairs (instead of just one pair of $q modes) [19] and keep only the on-resonance terms in the that are involved in this SORS process has the Hamiltonian. The off-resonance terms only con- form º(t)" qºq(t). Therefore, as long as the pho- tribute to virtual processes [20] at higher orders. ton depletion is negligible, all the phonon modes This approximation is appropriate for times much that are involved in a SORS process are driven into longer than the phonon period. The simplified two-mode quadrature squeezed states. In other 24 X. Hu, F. Nori / Physica B 263—264 (1999) 16—29 words, squeezing can be achieved in a continuum of then rotates the state by changing its phase [23]. phonon modes by a CW stimulated SORS process. The state will then freely evolve after t"0>. This result is consistent with Ref. [14] where the time- evolution operator is expressed in terms of real 8. Squeezed phonons via impulsive SORS phonon normal mode operators [18], instead of the complex ones used in this paper. Notice that, in Recently, an impulsive SORS process has been contrast to the CW SORS, the phase of the squeez- used to experimentally generate phonon squeezing ing factor f for the impulsive case is fixed by the [14]. Here we treat the problem expressing the time intensity of the light pulse. evolution operator of the system in terms of a product of the two-mode quadrature squeezing operator and the free rotation operators [23]. Since the incid- 9. Macroscopic implications and time-dependence ent photons are now in an ultrashort pulse, the com- of the dielectric constant plete Hamiltonian can be solved in the limit when the optical field can be represented by a d-function. Now that we have obtained the phonon states for Such an approximation is usually considered when both the CW and pulsed SORS cases, let us con- the optical pulse duration is much shorter than the sider the macroscopic implications of these states. optical phonon period, which is experimentally More generally, let us first discuss the implications feasible with femtosecond laser pulses. The Hamil- of the phonon squeezed states disregarding how tonian for the SORS can now be written as they are generated. An experimentally observable "+ + !jd , j H q Hq q (t)QqQ\q , where q carries the quantity O which is related to the atomic displace- information on the amplitudes of the incoming ments in the crystal can generally be expressed in + j j optical fields and the electronic polarizability. No- terms of Qq: O"O(0)# q( O/ Qq)Qq#2" j # # #2 " tice that the light-phonon coupling strength q in O O O where the first term O O(0) j the CW case has units of energy, while q here has is the operator O when all Qq’s vanish. An example units of . To further simplify the problem, we of an experimentally observable quantity O is the assume that only $q modes are involved in the change in the crystal dielectric constant de due to process. Such a simplification is possible when the the atomic displacement produced by the incident photon depletion and the phonon anharmonic in- electric fields. To first order in Qq , teraction are negligible, so that different pairs of j(de) W phonon modes are independent from each other. de"de " + (b #bR )e* q jQ 2u q \q The Hamiltonian is now OV q q W " !jd #(b #bR)e\* q]. Hq Hq q (t)QqQ\q, (37) \q q W j j "j de j and the Schro¨ dinger equation for these two-pho- Here q is the phase of O/ Qq ( )/ Qq. Indeed, j "t 2" "t 2 a widely used method to track the phases of coher- non modes is i R q(t) Hq q(t) . This equation can be solved by separating the free oscillator terms ent phonons in the time domain [2] is based on the and the two-phonon creation and annihilation observation of the reflectivity (or transmission) terms. The resulting time-dependent wave function modulation dR (d¹) of the sample, which is linearly is related to de — the change in the dielectric constant due to lattice vibrations. The above equation for de tH ijH "t t 2" q q q indicates that we can define a generalized [10,41] q( ) expG HexpG u H i q lattice amplitude operator [8,11]: u ($q)" W W E # R * q# # R \* q ; +fYH f R R ,"t \ 2 (bq b q)e (b q bq)e . This generalized exp q bqb q! qbqb q q(0 ) . (38) \ \ W \ \ + * q, lattice amplitude u ($q)"2Re Qqe is the un- E f j \*Hq Here q"!i qe / . Hence the effect of the derlying microscopic quantity related to an ob- optical pulse is clear: it first applies a two-mode served reflectivity or transmission modulation de quadrature squeezing operator on the initial state, when the linear term in Qq, , exists. X. Hu, F. Nori / Physica B 263—264 (1999) 16—29 25

1de 2 1D de 2 Even if vanishes, ( ) does not. Since has essentially the same form as the one obtained 1 2 1 2+ m different pairs of $q phonon modes are uncor- in Ref. [14]: Qq (t) " Qq (0) 1#2 q # de m u #u , u related to one another, the fluctuation of can be 2 q sin(2 qt q) . The small phase term q is ne- expressed as 1(*de )2"+ ( /2u )"j(de)/jQ " glected in [14] when computing transmission cha- OV q q 1* $ 2 "t 2" u uE( q) . Here the state is (t) nges. The difference in phases, rq versus q ,is "t 2 "t 2 º(t) (0) " qºq(t) q(0) in either the CW or the negligible in the limit of very small squeezing factor, impulsive case. We can again focus on a single pair and originates from the different interaction Hamil- of $q modes. In the CW case, using Eq. (36), the tonians used here and in Ref. [14]. The interaction $ fluctuation is term in Ref. [14] is proportional to uE( q) with W "0 (notice that their Q is real and based on 1*u($q)2(#)"2+e\Pq cos(X (t)# /2) q q E q standing wave quantization [18]). Therefore, the # Pq X # , e sin ( q(t) /2) , (39) interaction Hamiltonian in Ref. [14] is (in our nota- tion) »Ju($q)J2Q Q #Q#Q . How- where r ""f ""j t/ , X (t)"u t#p/4, and here- E q \q q \q q q q q q ever, the last two terms in this expression do not after 122 denotes an expectation value on satisfy momentum conservation, we thus did not squeezed states, unless stated otherwise. Therefore, include them and kept only Q Q in our interac- at certain times, the fluctuation 1*u($q)2(#) can q \q E tion term (this form is also used by Ref. [18]). be smaller than 2, which is the vacuum fluctuation When the linear perturbation de due to phonons level. Furthermore, all the pairs of phonon modes vanishes, such as in Ref. [14], the second order that are driven by the stimulated SORS process correction O ("de ) must be considered. When the share the same frequency: u "(u !u )/2. There- q phonon states are modulated by a SORS, so that fore, all the fluctuations 1*u($q)2(#) evolve with E the $q modes are the only ones which are corre- the same u . Notice that there is no dependence on q lated, de "+ (j(de)/jQ jQ )Q Q . Let us first W in the final expression of 1*u($q)2(#), and the q q \q q \q q E focus on one pair of $q modes in the CW case. In squeezing factor phase /2 has no q-dependence, a vacuum state, 10"Q Q "02"1, while in all the pairs of modes involved through the SORS q \q a squeezed vacuum state "02 , 10"Q Q "02 " share the same phase in their fluctuations. There- q \q 1*u($q)2(#)/2, with the right-hand side given in fore there can be squeezing in the overall fluctu- E Eq. (39). Therefore, the expectation value of Q Q ation 1(*de )2(#). Furthermore, the phase of this q \q in a squeezed vacuum state is periodically smaller overall fluctuation can be adjusted by tuning the than its vacuum state value. Let us now include all phase difference of the two incoming light beams. the phonon modes that contribute to de . In a vac- In the impulsive case [14,2,34], if the $q-mode uum state, 10"de "02"+ jde/(jQ jQ ). On the phonons are driven into a squeezed vacuum state, q q \q $ other hand, in a squeezed vacuum state, the fluctuation in uE( q)is 1*u($q)2"2+e\Pq cosX(t) 1 j(de) E q 1de 2" + 1*u($q)2(#). (41) j j E Pq X , 2 q Qq Q q #e sin q(t) , (40) \ ""f""j X "X ! where rq q q/ , and q(t) q(t) rq. Again, Since the phase /2 has no q-dependence, contri- the squeezing will reveal itself through oscillations butions from the phonon modes sharing the same 1 * de 2 in [ ( )] (q) which is proportional to frequency add up constructively. It is thus possible 1* $ 2 1de 2 uE( q) . Note that these oscillations are essen- that is periodically smaller than its vacuum 1de 2" tially the same as the ones obtained in the CW case. state value. Similarly, in the impulsive case, \+ j de j j 1* $ 2 However, now the squeezing factor is time-inde- 2 q( ( )/ Qq Q\q) uE( q) ; however, the " p ! 1de 2 pendent. Also, the t 0 phase /4 rq in Eq. (40) phase factor in has a q-dependence through is q-dependent. Eq. (40) can be rewritten as rq , so that all the phonon modes with the same 1* $ 2" + # u ! , u 1de 2 uE( q) 2 cosh 2rq sinh 2rq sin(2 qt rq) . q do not contribute to synchronously. In 1* $ 2" For small rq, this becomes uE( q) the CW SORS and in the very-small-rq limit impul- + Y u , 2 1#2rq #2rq sin(2 qt!rq) . This expression sive SORS the phase of the expectation value 26 X. Hu, F. Nori / Physica B 263—264 (1999) 16—29

1 2 u #u " QqQ\q does not depend on q; this is crucial to the frequencies of these modes satisfy q \q u !u experimental observation of modulations in the di- . The impulsive case is slightly different. electric constant, because this q-insensitivity leads Although the Hamiltonian is similar to a paramet- to constructive summations of all the q pairs in- ric process, the energy transfer from the photons to volved. Also, at a van Hove singularity a large the two phonon modes is instantaneous. The result- de number of modes contribute to with the same ing phonon state is a two-mode quadrature frequency and phase, thus their effect is larger and squeezed vacuum state. Indeed, a regular paramet- easier to observe [14]. ric process pumps energy into the signal and idler modes gradually, while the impulsive SORS does it suddenly. The correlation between the two phonon 10. Squeezed phonons via a finite-width SORS modes, and thus the squeezing effect, is also introduced instantaneously. Notice that this mechanism Real light pulses are not d-functions. Therefore, is reminiscent of the frequency-jump mechanism we have also considered a SORS pumped by a light proposed in [6,37,38]. In the impulsive SORS, the pulse with a finite width (smaller than the phonon frequency of the phonon modes has an ‘infinite’ period ¹) instead of a d-function. For a fixed peak d-peak change at t"0, while the frequency-jump height I, we find [22] that the optimal pulse width mechanism has finite frequency changes, and ¹ that maximizes the squeezing effect satisfies squeezing there can be intensified by repeated fre- ¹+¹ /4.4. This calculation indicates that the ex- quency jumps at appropriate times. However, as it periments [14] used a pulse width which is near the has been pointed out in [6,37,38], a finite frequency ¹ + + +¹ optimal value ( /4.4 300/4.4 fs 68 fs ). jump up immediately followed by an equal jump The calculation [22] can be summarized as follows. down results in no squeezing at all. First, in the impulsive Hamiltonian we replace the d ¹ -function by a Gaussian with its width as a variational parameter. Since now the Hamil- 12. Experimental search for squeezed phonons tonian is time-dependent in the interaction picture, we cannot directly integrate the Schro¨ dinger equa- One group [14] has claimed to detect squeezed tion. Instead, we use the Magnus method to obtain phonons. However, no other group has succeeded the time evolution operator and keep only the in reproducing their results. Recent calculations dominant first term. This approximation is valid indicate that these initial experiments [14] do not when the pulse duration is shorter than the phonon prove that phonon squeezing has been achieved ¹ period. We then calculate the width of the experimentally. Here we discuss several important Gaussian that maximizes the squeezing factor. For points to be considered for the eventual future ex- a constant peak intensity, a pulse that is too narrow perimental observation of a phonon squeezed state. does not contain enough photons; while it can be First of all, squeezing must refer to a phonon proven that a pulse which is too long (i.e., with mode with a variance that falls below the standard a width comparable to ¹), attenuates the squeezing quantum limit. This crucial comparison has not effect. been produced yet. Thus, a reliable determination of the vacuum noise level is necessary in order to establish if a squeezing of the vacuum noise has 11. Phonon squeezing mechanism indeed occurred. Experiments in quantum optics have an independent way to reliably obtain the What is the mechanism of phonon squeezing in noise level of the vacuum: by using an independent the SORS processes? For the CW case, the Hamil- light beam as a local oscillator. Currently, there is tonian is the same as an optical two-mode paramet- no such phase-sensitive detection scheme for pho- ric process [19], with the low frequency interference nons. However, as we point out later, a pump- of the combined photon modes as the pump, the probe experiment may be able to establish such two phonon modes as the signal and idler. The a criterion, but discretion has to be applied. X. Hu, F. Nori / Physica B 263—264 (1999) 16—29 27

Second, for phonons, with relatively low energy through an impulsive SORS process in a pump- compared to photons, thermal noise should always probe experiment. We then give a numerical esti- be considered in an experiment. This is especially mate on whether squeezing of quantum noise has important when the noise modulation factor is been achieved in [14]. small, such as the case in the reported experiment Experimentally, the observed quantity is the [14], where the noise modulation factor is only change in the transmission ¹ due to the impulsive

0.0001%. Achieving total noise modulation, as can SORS process. Up to second order in Qq, ¹ can be be done through CW and impulsive SORS, is not expressed as equivalent to the squeezing of the vacuum noise. j¹ j¹ Only when the noise modulation factor is big ¹"¹ #+ 1Q 2#+ 1Q Q2 j q j j q q , enough to overcome the thermal noise, then the q Qq q Qq Qq quantum noise is suppressed. where the average is over the phonon states of the Third, in that experiment, the squeezing factor crystal. In a squeezed vacuum (or thermal) state is obtained from the modulation of the light trans- considered here, 1Q 2"0 and 1Q Q 2" mission through a crystal. However, this change in q q \q 21*u($q, t)2. Therefore, ¹ does contain informa- light transmission describes the modulation of the tions on the atomic lattice fluctuations. total fluctuations (both quantum and thermal) of Through impulsive SORS, we can achieve modu- the atomic displacements. Thus, by itself this lation of the total noise of the system, and the effect modulation of the noise is not a proof of quantum can be seen through a modulation of the transmis- noise suppression, let alone of squeezing below the sion ¹, or more specifically, *¹/¹. Starting from vacuum noise level. a thermal state, we have Fourth, the efficiency of the signal detection is a crucial issue in photon squeezing experiments, *¹ 1 j¹ + + # N u r u t but not addressed in its phonon ‘analog’ experi- ¹ ¹ j j [1 2 ( q)] q sin(2 q ). q Qq Q\q ment. In a homodyne or heterodyne detector for photons, if the efficiency of the photon counters is Notice that this modulation of the total noise is not less than one, additional noise is introduced into equivalent to a squeezing of quantum noise, be- the signal. Similarly, in a pump-probe phonon de- cause *¹"0 is an indication of total noise limit tection scheme, the probe light pulse and photo- (that is, quantum noise plus thermal noise), not detector should introduce additional noise into the quantum noise limit. Even though the experiment is final signal. Therefore, a careful analysis of these done at low temperatures, so that thermal noise is additional noise sources should be performed when miniscule, we still have to compare the thermal employing a pump-probe scheme to detect phonon noise to the squeezing factor (or modulation factor squeezing. through impulsive SORS) because the later is very For the above reasons, it is premature and un- small, too. One criteria for quantum noise sup- warranted to claim that squeezed phonons have pression would be been observed experimentally. The evidence pre- *¹ ¹!¹ sented so far is incomplete and inconclusive. The " !# ¹ ¹ answer to the question: ‘Can phonons be squeezed !# like photons?’ is yes on theoretical grounds, but the 1 j¹ + + [2N(u )#r sin(2u t)]. experimental proof still lies in the future. ¹ jQ jQ q q q The rest of this section presents quantitative deri- !# q q \q vations in support of the statements made above. A negative *¹/¹ here indicates that, at least in Even with the reservations presented above regard- some spectral regions, the atomic displacement ing the pump-probe scheme, we would like to point noise is squeezed below the ground state limit. We u out that it does provide a possible means for the are now comparing the thermal population 2N( q) detection of phonon squeezing. Here we discuss the and the squeezing factor rq. In summary, there is criteria for achieving squeezing of quantum noise always modulation of total noise by the impulsive 28 X. Hu, F. Nori / Physica B 263—264 (1999) 16—29

u SORS, but only when rq'2N( q) do we achieve terms of the phonon period, that maximizes the noise smaller than the ground state limit. squeezing effect. For example, at an experimental temperature of ¹ 10 K, the corresponding thermal energy k is about 0.88 meV. The acoustic phonons involved Acknowledgements in the experiment reported in Ref. [14] have a frequency of about 2.7 THz, which corresponds to We acknowledge useful conversations with S. an energy quantum of u+10.6 meV. Therefore, Hebboul, S. Tamura and H. Wang. One of us (XH) the thermal noise factor here is about N" acknowledges support from the US Army Research + u ¹ ! ,+ \+ ; \ Office. 1/ exp( /k ) 1 e 0.6 10 , which is almost identical to the squeezing factor given in Ref. [14]. Therefore, it is quite clear that this particular experiment has not achieved squeezing of References quantum noise below the vacuum noise limit. How- ever, if the experimental temperature is further [1] V. Narayanamurti et al., Phys. Rev. Lett. 43 (1979) 2012. [2] S. Ruhman, A.G. Joly, K. Nelson, IEEE J. Quantum Elec- lowered, the thermal factor will become smaller. tron. 24 (1988) 460. The squeezing action initiated by the second-order [3] F. Nori, R. Merlin, S. Haas, A.W. Sandvik, E. Dagotto, Raman scattering should then be strong enough to Phys. Rev. Lett. 75 (1995) 553. realize suppression of noise below the ground state [4] Appl. Phys. B 55 (3) 1992. level. [5] J. Sidles, D. Rugar, Phys. Rev. Lett. 70 (1993) 3506. [6] J. Janszky, P. Adam, Phys. Rev. A 46 (1992) 6091. [7] M. Artoni, J.L. Birman, Opt. Commun. 89 (1992) 324. [8] X. Hu, F. Nori, Phys. Rev. B 53 (1996) 2419. 13. Conclusions [9] X. Hu, F. Nori, Bull. Am. Phys. Soc. 39 (1994) 466. [10] X. Hu, F. Nori, 1995, UM preprint 8-15-95. We have presented an overview of the definitions [11] X. Hu, F. Nori, Phys. Rev. Lett. 76 (1996) 2294. [12] X. Hu, F. Nori, Phys. Rev. Lett. 79 (1997) 4605. relevant to quantum phonon optics, including quan- [13] P.F. Schewe, B. Stein, AIP Phys. News 261, 6 March, 1996. tum coherent and squeezed phonons. Afterwards, [14] G. Garett et al., Science 275 (1997) 1638. we have studied theoretically the generation of [15] W. Kohn, D. Sherrington, Rev. Mod. Phys. 42 (1970) 1. phonon squeezing using a stimulated SORS pro- [16] C.W. Gardiner, Quantum Noise, Springer, Berlin, 1991. cess. In particular, we calculated the time evolution [17] D.F. Walls, Nature 306 (1983) 141. [18] M. Born, K. Huang, Dynamical Theory of Crystal Latti- operators of the phonons in two different cases: ces, Oxford University Press, Oxford, 1954. when the incident photons are in monochromatic [19] M. Schubert, B. Wilhelmi, Nonlinear Optics and Quantum continuous waves, and when they are in an ultra- Electronics, Wiley, New York, 1986. short pulse. The amplitude of the squeezing factor [20] G.P. Srivastava, The Physics of Phonons, Hilger, Philadel- initially increases with time and then saturates in phia, 1990. [21] R. Loudon, P.L. Knight, J. Mod. Opt. 34 (1987) 709. the CW SORS case, while it remains constant in the [22] X. Hu, F. Nori, unpublished. pulsed SORS case. In addition, the t"0 phase of [23] B.L. Schumaker, Phys. Rep. 135 (1986) 317. the squeezing factor in the CW SORS, , can [24] C.F. Quate, Sci. Am. 241 (1979) 62. be continuously adjusted by tuning the relative [25] J.P. Wolfe, Phys. Today 33 (12) (1980) 44. phase of the two incoming monochromatic photon [26] V. Narayanamurti, Science 213 (1981) 717. Jj [27] S. Tamura, Phys. Rev. B 30 (1984) 849. beams, while for the pulsed SORS the phase ( q) [28] G.A. Northrop, S.E. Hebboul, J.P. Wolfe, Phys. Rev. Lett. of the squeezing factor is determined by the ampli- 55 (1985) 95. tude of the incoming light pulse. For both cases we [29] S. Tamura, F. Nori, Phys. Rev. B 38 (1988) 5610. calculated the quantum fluctuations of a generaliz- [30] S.E. Hebboul, J.P. Wolfe, Z. Phys. B 73 (1989) 437. S. ed lattice amplitude operator and the second order Tamura, in: S. Hunkulinger, W. Ludwig, G. Weiss (Eds.), Phonon 89, World Scientific, Singapore, 1989, p. 703. contribution to the change in dielectric constant, [31] S. Tamura, F. Nori, Phys. Rev. B 41 (1990) 7941. which is measurable. For the finite-width impulsive [32] N. Nishiguchi, S. Tamura, F. Nori, Phys. Rev. B 48 (1993) case, we computed the optimal pulse width, in 2515. X. Hu, F. Nori / Physica B 263—264 (1999) 16—29 29

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