Quantum Phonon Optics: Coherent and Squeezed Atomic Displacements

PHYSICAL REVIEW B VOLUME 53, NUMBER 5 1 FEBRUARY 1996-I

Quantum phonon optics: Coherent and squeezed atomic displacements

Xuedong Hu and Franco Nori Department of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120 ͑Received 17 August 1995; revised manuscript received 27 September 1995͒ We investigate coherent and squeezed quantum states of phonons. The latter allow the possibility of modu- lating the quantum ﬂuctuations of atomic displacements below the zero-point quantum noise level of coherent states. The expectation values and quantum ﬂuctuations of both the atomic displacement and the lattice amplitude operators are calculated in these states—in some cases analytically. We also study the possibility of squeezing quantum noise in the atomic displacement using a polariton-based approach.

I. INTRODUCTION words, a coherent state is as ‘‘quiet’’ as the vacuum state. Squeezed states5 are interesting because they can have Classical phonon optics1 has succeeded in producing smaller quantum noise than the vacuum state in one of the many acoustic analogs of classical optics, such as phonon conjugate variables, thus having a promising future in differ- mirrors, phonon lenses, phonon filters, and even ‘‘phonon ent applications ranging from gravitational wave detection to microscopes’’ that can generate acoustic pictures with a reso- optical communications. In addition, squeezed states form an lution comparable to that of visible light microscopy. Most exciting group of states and can provide unique insight into phonon optics experiments use heat pulses or superconduct- quantum mechanical fluctuations. Indeed, squeezed states are ing transducers to generate incoherent phonons, which now being explored in a variety of non-quantum-optics sys- 6 propagate ballistically in the crystal. These ballistic incoher- tems, including classical squeezed states. ent phonons can then be manipulated by the above- In Sec. II we introduce some quantities of interest and mentioned devices, just as in geometric optics. study the fluctuation properties of the phonon vacuum and Phonons can also be excited phase coherently. For in- number states. In Secs. III and IV we investigate phonon stance, coherent acoustic waves with frequencies of up to coherent and squeezed states. In Sec. V we propose a way of 1010 Hz can be generated by piezoelectric oscillators. Lasers squeezing quantum noise in the atomic displacement opera- have also been used to generate coherent acoustic and optical tor using a polariton-based mechanism. The Appendix sum- phonons through stimulated Brillouin and Raman scattering marizes the derivation of the time evolution of the relevant experiments. Furthermore, in recent years, it has been pos- operators in this polariton approach. Finally, Sec. VI presents sible to track the phases of coherent optical phonons,2 due to some concluding remarks. the availability of femtosecond-pulse ultrafast lasers ͑with a 3 pulse duration shorter than a phonon period͒, and techniques II. PHONON OPERATORS AND THE PHONON VACUUM that can measure optical reflectivity with accuracy of one AND NUMBER STATES part in 106. In most situations involving phonons, a classical descrip- A phonon with quasimomentum pϭបq and branch sub- tion is adequate. However, at low enough temperatures, script ␭ has energy ⑀q␭ϭប␻q␭ ; the corresponding creation quantum fluctuations become dominant. For example, a re- and annihilation operators satisfy the boson commutation recent study4 shows that quantum fluctuations in the atomic lations positions can indeed influence observable quantities ͑e.g., the † Raman line shape͒ even when temperatures are not very low. ͓bqЈ␭Ј,bq␭͔ϭ␦qqЈ␦␭␭Ј, ͓bq␭ ,bqЈ␭Ј͔ϭ0. ͑1͒ With these facts in mind, and prompted by the many exciting developments in classical phonon optics, coherent phonon The atomic displacements ui␣ of a crystal lattice are given by experiments, and ͑on the other hand͒ squeezed states of 5 light, we would like to explore phonon analogs of quantum 1 N ␭ ␭ iq•Ri optics. In particular, we study the dynamical and quantum ui␣ϭ ͚ Uq␣Qqe . ͑2͒ fluctuation properties of the atomic displacements, in anal- ͱNm q␭ ogy with the modulation of quantum noise in light. Specifi- Here Ri refers to the equilibrium lattice positions, ␣ to a cally, we study single-mode and two-mode phonon coherent ␭ and squeezed states, and then focus on a polariton-based ap- particular direction, and Qq is the normal-mode amplitude proach to achieve smaller quantum noise than the zero-point operator fluctuations of the atomic lattice. The concepts of coherent and squeezed states were both ␭ ប † Qqϭͱ ͑bq␭ϩbϪq␭͒. ͑3͒ proposed in the context of quantum optics. A coherent state 2␻q␭ is a phase-coherent sum of number states. In it, the quantum fluctuations in any pair of conjugate variables are at the An experimentally observable quantity is the real part of the lower limit of the Heisenberg uncertainty principle. In other Fourier transform of the atomic displacement:

ប noiseless as the vacuum state. Coherent states are also the set Re͓u ͑q͔͒ϭ ͕U␭ ͑b ϩb† ͒ of quantum states that best describe the classical harmonic ␣ ͚ ͱ8m␻ q␣ q␭ Ϫq␭ ␭ q␭ oscillators.7 ␭ † A single-mode phonon coherent state can be generated by ϩU *͑bϪq ϩb ͖͒. ͑4͒ q␣ ␭ q␭ the Hamiltonian For simplicity, hereafter we will drop the branch subscript ␭, assume that U is real, and define a q-mode dimension- 1 q␣ Hϭប␻ b†b ϩ ϩ␭*͑t͒b ϩ␭ ͑t͒b† ͑14͒ less lattice amplitude operator: qͩ q q 2ͪ q q q q † † u͑Ϯq͒ϭbqϩbϪqϩbϪqϩbq . ͑5͒ and an appropriate initial state. Here ␭q(t) represents the interaction strength between phonons and the external This operator contains essential information on the lattice source. More specifically, if the initial state is a vacuum dynamics, including quantum fluctuations. It is the phonon state, ͉␺(0)͘ϭ͉0͘, then the state vector becomes a single- analog of the electric field in the photon case. mode coherent state thereafter, Let us first consider the phonon vacuum state. When no Ϫi␻qt phonon is excited, the crystal lattice is in the phonon vacuum ͉␺͑t͒͘ϭ͉⌳q͑t͒e ͘, ͑15͒ state ͉0͘. The expectation values of the atomic displacement and the lattice amplitude are zero, but the fluctuations will be where finite: i t ⌳ t ϭϪ ␭ ␶ ei␻q␶d␶ ͑16͒ 2 2 2 q͑ ͒ ͵ q͑ ͒ ͗͑⌬ui␣͒ ͘vacϵ͗͑ui␣͒ ͘vacϪ͗ui␣͘ vac ͑6͒ ប Ϫϱ

N 2 is the coherent amplitude of mode q. If the initial state is a ប͉Uq␣͉ ϭ͚ , ͑7͒ single-mode coherent state ͉␺(0)͘ϭ͉␣q͘, then the state vec- q 2Nm␻q␣ tor at time t takes the form

2 Ϫi␻qt ͓͗⌬u͑Ϯq͔͒ ͘vacϭ2. ͑8͒ ͉␺͑t͒͘ϭ͉͕⌳q͑t͒ϩ␣q͖e ͘, ͑17͒ Let us now consider the phonon number states. The eigen- which is still a coherent state. Ϫi␻qt states of the harmonic phonon Hamiltonian are number states In a single-mode (q) coherent state ͉⌳q(t)e ͘, ͗ui␣(t)͘ coh and ͗u(Ϯq)͘coh are sinusoidal functions of time. which satisfy bq͉nq͘ϭͱnq͉nqϪ1͘. The phonon number and the phase of atomic vibrations are conjugate variables. Thus, The fluctuation in the atomic displacements is due to the uncertainty principle, the phase is arbitrary when N ប͉U ͉2 the phonon number is certain, as is the case with any number 2 q␣ ͗͑⌬ui␣͒ ͘cohϭ͚ . ͑18͒ state ͉nq͘. Thus, in a number state, the expectation values of q 2Nm␻q␣ the atomic displacement n u n and q-mode lattice am- ͗ q͉ i␣͉ q͘ The unexcited modes are in the vacuum state and thus all plitude n u(Ϯq) n vanish due to the randomness in the ͗ q͉ ͉ q͘ contribute to the noise in the form of zero-point fluctuations. phase of the atomic displacements. The fluctuations in a Furthermore, number state ͉nq͘ are ͓͗⌬u͑Ϯq͔͒2͘ ϭ2. ͑19͒ U 2n N U 2 coh 2 ប͉ q␣͉ q ប͉ qЈ␣͉ ͗͑⌬u ͒ ͘ ϭ ϩ , ͑9͒ 2 i␣ num Nm␻ ͚ 2Nm␻ From the expressions of the noise ͗(⌬ui␣) ͘coh and q␣ qЈÞq qЈ␣ 2 ͓͗⌬u(Ϯq)͔ ͘coh , it is impossible to know which state ͑if ͓͗⌬u͑Ϯq͔͒2͘ ϭ2ϩ2n . ͑10͒ any͒ has been excited, while this information is clearly num q present in the expression of the expectation value of the lattice amplitude ͗u(Ϯq)͘coh . These results can be straightfor- III. PHONON COHERENT STATES wardly generalized to multimode coherent states. A single-mode (q) phonon coherent state is an eigenstate of a phonon annihilation operator: IV. PHONON SQUEEZED STATES In order to reduce quantum noise to a level below the bq͉␤q͘ϭ␤q͉␤q͘. ͑11͒ zero-point fluctuation level, we need to consider phonon It can also be generated by applying a phonon displacement squeezed states. Quadrature squeezed states are generalized 8 operator Dq(␤q) to the phonon vacuum state coherent states. Here ‘‘quadrature’’ refers to the dimensionless coordinate and momentum. Compared to coherent states, † * ͉␤q͘ϭDq͑␤q͉͒0͘ϭexp͑␤qbqϪ␤q bq͉͒0͘ ͑12͒ squeezed ones can achieve smaller variances for one of the quadratures during certain time intervals and are therefore 2 ϱ nq ͉␤q͉ ␤q helpful for decreasing quantum noise. Figures 1 and 2 sche- ϭexp Ϫ ͚ ͉nq͘. ͑13͒ matically illustrate several types of phonon states, including ͩ 2 ͪ n ϭ0 q ͱn ! q vacuum, number, coherent, and squeezed states. These fig- Thus it can be seen that a phonon coherent state is a phase- ures are the phonon analogs of the illuminating schematic coherent superposition of number states. Moreover, coherent diagrams used for photons.8 states are a set of minimum-uncertainty states which are as A single-mode quadrature phonon squeezed state is gen- 53 QUANTUM PHONON OPTICS: COHERENT AND SQUEEZED . . . 2421

FIG. 2. Schematic diagram of the time evolution of the expectation value and the ﬂuctuation of the lattice amplitude operator u(Ϯq) in different states. Dashed lines represent ͗u(Ϯq)͘, while the solid lines represent the envelopes ͗u(Ϯq)͘Ϯͱ͓͗⌬u(Ϯq)͔2͘. ͑a͒ The phonon vacuum state ͉0͘, where ͗u(Ϯq)͘ϭ0 and 2 ͓͗⌬u(Ϯq)͔ ͘ϭ2. ͑b͒ A phonon number state ͉nq ,nϪq‹, where 2 ͗u(Ϯq)͘ϭ0 and ͓͗⌬u(Ϯq)͔ ͘ϭ2(nqϩnϪq)ϩ2. ͑c͒ A single- mode phonon coherent state ͉␣q͘, where ͗u(Ϯq)͘ϭ2͉␣q͉cos␻qt 2 ͑i.e., ␣q is real͒, and ͓͗⌬u(Ϯq)͔ ͘ϭ2. ͑d͒ A single-mode phonon squeezed state ͉␣ eϪi␻qt,␰(t)͘, with the squeezing factor FIG. 1. Schematic diagram of the uncertainty areas ͑shaded͒ in q ␰(t)ϭreϪ2i␻qt and rϭ1. Here, ͗u(Ϯq)͘ϭ2͉␣ ͉cos␻ t, and the generalized coordinate and momentum X(q,Ϫq), P(q,Ϫq) q q „ … ͓͗⌬u(Ϯq)͔2͘ϭ2(eϪ2rcos2␻ tϩe2rsin2␻ t). ͑e͒ A single-mode phase space of ͑a͒ the phonon vacuum state, ͑b͒ a phonon number q q phonon squeezed state, as in ͑d͒; now the expectation value of u is state, ͑c͒ a phonon coherent state, and ͑d͒ a phonon squeezed state. ͗u(Ϯq)͘ϭ2͉␣ ͉sin␻ t ͑i.e., ␣ is purely imaginary͒, and the ﬂuc- Here X(q,Ϫq and P(q,Ϫq are the two-mode (Ϯq) coordinate q q q … … tuation ͓͗⌬u(Ϯq)͔2͘ has the same time dependence as in ͑d͒. No- and momentum operators, which are the direct generalizations of tice that the squeezing effect now appears at the times when the their corresponding single-mode operators. Notice that the phonon lattice amplitude ͗u(Ϯq)͘ reaches its maxima, while in ͑d͒ the coherent state has the same uncertainty area as the vacuum state, squeezing effect is present at the times when ͗u(Ϯq)͘ is close to and that both areas are circular, while the squeezed state has an zero. elliptical uncertainty area. Therefore, in the direction parallel to the ␪/2 line, the squeezed state has a smaller noise than both the H ϭប␻ b† b ϩប␻ b† b ϩ␨͑t͒b† b† vacuum and coherent states. q1 ,q2 q1 q1 q1 q2 q2 q2 q1 q2 ϩ␨*͑t͒b b . ͑24͒ erated from a vacuum state as q1 q2 The time-evolution operator has the form ͉␣q ,␰͘ϭDq͑␣q͒Sq͑␰͉͒0͘; ͑20͒ a two-mode quadrature phonon squeezed state is generated i † † U͑t͒ϭexp Ϫ H0t exp͓␰*͑t͒bq bq Ϫ␰͑t͒b b ͔, as ͩ ប ͪ 1 2 q1 q2 ͑25͒ ͉␣ ,␣ ,␰͘ϭD ͑␣ ͒D ͑␣ ͒S ͑␰͉͒0͘. ͑21͒ q1 q2 q1 q1 q2 q2 q1 ,q2 where

Here Dq(␣q) is the coherent state displacement operator with † † i ␾ H0ϭប␻q bq bq ϩប␻q bq bq ͑26͒ ␣qϭ͉␣q͉e , 1 1 1 2 2 2 and ␰* 2 ␰ †2 Sq͑␰͒ϭexp bqϪ bq , ͑22͒ ͩ 2 2 ͪ i t i͑␻ ϩ␻ ͒␶ ␰͑t͒ϭ ␨͑␶͒e q1 q2 d␶. ͑27͒ ប ͵Ϫϱ S ͑␰͒ϭexp͑␰*b b Ϫ␰b† b† ͒ ͑23͒ q1 ,q2 q1 q2 q1 q2 Here ␰(t) is the squeezing factor and ␨(t) is the strength of are the single- and two-mode squeezing operators, and the interaction between the phonon system and the external i␪ ␰ϭre is the complex squeezing factor with rу0 and source; this interaction allows the generation and absorption 0р␪Ͻ2␲. The squeezing operator S (␰) can be pro- q1 ,q2 of two phonons at a time. The two-mode phonon quadrature duced by the following Hamiltonian: operators have the form 2422 XUEDONG HU AND FRANCO NORI 53

Ϫ3/2 † † TABLE I. Different combinations of tϭ0 initial states ͑modes X͑q,Ϫq͒ϭ2 ͑bqϩbqϩbϪqϩbϪq͒ ͑28͒ Ϯk) for the polariton approach to lattice amplitude squeezing and ϭ2Ϫ3/2u͑Ϯq͒, ͑29͒ the corresponding effects in the fluctuations of the lattice amplitude operator u(Ϯk). Here CS(k), VS(k), TS(k), SMST(k), and TMST P͑q,Ϫq͒ϭϪi2Ϫ3/2͑b Ϫb†ϩb Ϫb† ͒. ͑30͒ (Ϯk) refer, respectively, to coherent, vacuum, thermal, single- q q Ϫq Ϫq mode, and two-mode squeezed states in the mode inside the paren- We have considered two cases where squeezed states theses, k or Ϯk. Ts(␹) is the temperature below which squeezing is were involved in modes Ϯq. In the first case, the system is obtained ͑see Fig. 4͒. By squeezing we mean that the quantum noise in a two-mode (Ϯq) squeezed state ͉␣ ,␣ ,␰ , of the relevant variable is below its corresponding vacuum state q Ϫq ‹ value. (␰ϭrei␪), and its fluctuation is

␪ ␪ tϭ0 photons tϭ0 phonons Squeezed u(Ϯk)? ͓͗⌬u͑Ϯq͔͒2͘ϭ2 eϪ2r cos2 ϩe2r sin2 . ͑31͒ ͩ 2 2ͪ CS(Ϯk) CS(Ϯk) yes ͑no͒ if ␹Ͼ(р) 0.1 SMST(k), VS(Ϫk VS(Ϯk) yes ͑no͒ if ␹Ͼ(р) 0.1 In the second case, the system is in a single-mode squeezed … SMST(k), VS(Ϫk TS(Ϯk) yes if TϽT (␹) state ͉␣ ,␰͘ (␣ ϭ͉␣ ͉ei␾) in the first mode and an arbitrary … s q q q TMST(Ϯk) VS(Ϯk) weak ͑no͒ if ␹Ͼ(р) 0.1 coherent state ͉␤Ϫq‹ in the second mode. The fluctuation is now H , the two free oscillator sums correspond to free ␪ ␪ polariton ͓͗⌬u͑Ϯq͔͒2͘ϭ1ϩe2r sin2 ␾ϩ ϩeϪ2r cos2 ␾ϩ . photons and free phonons, while the mixing terms come ͩ 2ͪ ͩ 2ͪ from the interaction E•P between photons and phonons. The ͑32͒ phonon energy E2k has been corrected as ␻0 is substituted In both of these cases, ͓͗⌬u(Ϯq)͔2͘ can be smaller than in by ␻0ͱ1ϩ␹, so that we have ‘‘dressed’’ phonons. Our goal is to compute the fluctuations of the lattice am- coherent states ͑see Fig. 2͒. † † plitude operator u(Ϯk,t)ϭbk(t)ϩbϪk(t)ϩbϪk(t)ϩbk(t). In a two-mode (Ϯk) coherent state ͉␣q ,␣Ϫq͘, its variance is V. POLARITON APPROACH 2 ͓͗⌬u(Ϯq)͔ ͘cohϭ2. Therefore, if at any given time we ob- Phonon squeezed states can be generated through phonon- tain a value less than 2, the lattice amplitude of the relevant phonon interactions. This will be discussed elsewhere.9 Here mode is squeezed. In our calculation, we diagonalize the po- we focus on how to squeeze quantum noise in the atomic lariton Hamiltonian and find the time dependence of displacements through phonon-photon interactions. When an u(Ϯq). The Appendix presents in more detail the derivation ionic crystal is illuminated by light, there can be a strong of the time evolution of u(Ϯq). coupling between photons and the local polarization of the Our results show that the fluctuation property of u(Ϯq) crystal in the form of phonons. Photons and phonons with sensitively depends on the tϭ0 initial state ͉␺(0)͘ of both the same wave vector can thus form polaritons.10 Although phonons and photons. Our results are summarized in Table I, now phonons and photons are not separable in a polariton, and some numerical examples are shown in Fig. 3. These we can still study the quantum noise in the atomic displace- calculations focus on the case where ck is close to ␻0 ͑the bare phonon frequency, which is typically 10 THz for op- ments. Let us consider the simplest Hamiltonian10 describing ϳ tical phonons͒ and thus our typical time is ϳ0.1 ps. More the above scenario: specifically, squeezing effects in u(Ϯk) are relatively strong for either one of the following two sets of tϭ0 initial states: † † † † Hpolaritonϭ͚ ͕E1kakakϩE2kbkbkϩE3k͑akbkϪakbk ͑i͒ photon and phonon coherent states, or ͑ii͒ single-mode k photon squeezed state and phonon vacuum state. For in- stance, the maximum squeezing exponent r is 0.015 when Ϫa b ϩa† b†͒ , ͑33͒ k Ϫk Ϫk k ͖ the incident photon state has a squeezing factor where ␰ϭ0.1e2ickt ͑where ck is the photon frequency͒.Onthe other hand, with an initial two-mode photon (Ϯk) squeezed E1kϭបck, ͑34͒ state and two-mode (Ϯk) phonon vacuum state, the squeezing effect in u(Ϯk) is weak. We have also used initial con- E2kϭប␻0ͱ1ϩ␹, ͑35͒ ditions with a single-mode photon squeezed state and thermal states in the two phonon modes. 2 1/2 ប ck␻0␹ Figure 4 shows the temperature dependence of the E3kϭi . ͑36͒ ͩ 4ͱ1ϩ␹ ͪ squeezing effect for several values of the dielectric susceptibility ␹ of the crystal. Our numerical results show that Here k is the wave vector for both photons and phonons and squeezing effects are quickly overshadowed by the thermal ␻0 is the bare phonon frequency. ␹ is the dimensionless noise for small ␹, while for large ␹ ͑e.g., ␹ϭ0.5) the dielectric susceptibility of the crystal ͑the strength of the squeezing effect can exist up to TϷ250 K, as illustrated in phonon-photon interaction͒ defined by Fig. 4. 2 ¨ 2 ␹␻0␧0EϭPϩ␻0P, ͑37͒ VI. CONCLUSIONS where E is the electric field of the incoming light and P is the In conclusion, we have investigated the dynamics and polarization generated by optical phonons in the crystal. In quantum fluctuation properties of phonon coherent and 53 QUANTUM PHONON OPTICS: COHERENT AND SQUEEZED . . . 2423

2 FIG. 3. Calculated ͓͗⌬u(Ϯk)͔ ͘ versus time for different com- FIG. 4. Temperature dependence of the minimum fluctuation binations of photon and phonon initial states using a polariton min͕͓͗⌬u(Ϯk)͔2͖͘ in u(Ϯk) using a polariton mechanism for mechanism for lattice amplitude squeezing. Dashed ͑solid͒ lines squeezing. The phonon frequency is 10 THz. The initial states are a correspond to a susceptibility ␹ϭ0.1(0.4). Time is measured in single-mode squeezed state in photon mode k, vacuum state in units of 1/ck, where ck is the free photon frequency. These calcu- photon mode Ϫk, and thermal state in both (Ϯk) phonon modes. 2it lations focus on the case where ck is close to ␻0 ͑the bare phonon The squeezing factor is ␰ϭ0.1e . Squeezing can exist up to a frequency, which is typically ϳ10 THz for optical phonons͒ and temperature Ts(␹). For example, when ␹ϭ0.2, squeezing effects thus our typical time is ϳ0.1 ps. The horizontal lines at vanish when Tտ25 K. On the other hand, for stronger photon- 2 ͓͗⌬u(Ϯk)͔ ͘ϭ2 correspond to the noise level of coherent states. phonon interaction ͑e.g., ␹ϭ0.5), the squeezing effects can be ob- 2 Thus, any time the fluctuation satisfies ͓͗⌬u(Ϯk)͔ ͘Ͻ2 ͑high- tained up to TϷ250 K. lighted͒, the state is squeezed. Different combinations of initial states were considered. ͑a͒ Photon and phonon coherent states. ͑b͒ mental developments in the still unexplored area of quantum Single-mode squeezed state in photon mode k with squeezing factor phonon optics and the manipulation of phonon quantum fluc- ␰ϭ0.1 and a vacuum state in the photon mode Ϫk; both phonon tuations. modes are in the vacuum state. ͑c͒ Same combination of states as in ͑b͒, but here ␰ϭ0.1e2it. ACKNOWLEDGMENTS squeezed states. In particular, we calculate the experimentally observable time evolution and fluctuation of the lattice It is a great pleasure for us to acknowledge useful conver- sations with Saad Hebboul, Roberto Merlin, Nicolas Bon- amplitude operator u(Ϯq). We show that the ͗u(Ϯq)͘ are adeo, Hailin Wang, Duncan Steel, Jeff Siegel, and especially sinusoidal functions of time in both coherent and squeezed Shin-Ichiro Tamura. states, but the fluctuation ͓͗⌬u(Ϯq)͔2͘ in a squeezed phonon state is periodically smaller than its vacuum or coherent state value 2. Therefore phonon squeezed states are periodi- APPENDIX: TIME EVOLUTION OF THE PHONON cally quieter than the vacuum state. In the polariton approach OPERATORS IN A POLARITON to squeezing, we calculate the atomic displacement part of a To derive the time evolution of the phonon operators, we polariton, and prove that the fluctuations of the associated need to first diagonalize the polariton Hamiltonian lattice amplitude operator can be squeezed for different com- H . For this purpose, we introduce the polariton opera- binations of initial photon and phonon states and large polariton tors ␣ in terms of the phonon and photon operators b and enough ( Ͼ ik k ␹ 0.1) interaction strength. a : It is difficult to generate squeezed states because they k have noise levels which are even lower than the one for the ␣ ϭw a ϩx b ϩy a† ϩz b† , iϭ1,2. ͑A1͒ vacuum state. Indeed, the experimental and theoretical devel- ik i k i k i Ϫk i Ϫk opment of photon coherent and squeezed states took decades. If we write Likewise, the experimental realization of phonon squeezed † † T states might require years of further theoretical and experi- ␣ϭ͑␣1k ,␣2k ,␣1,Ϫk ,␣2,Ϫk͒ , ͑A2͒ mental work. Nevertheless, we believe that theoretical results in quantum phonon optics can help the development of the † † T aϭ͑ak ,bk ,aϪ ,bϪ ͒ , ͑A3͒ corresponding experiments. We hope that our effort9 into this k k very rich problem will lead to more theoretical and experi- the above relation can be written in a matrix form 2424 XUEDONG HU AND FRANCO NORI 53

which is true if the two different polariton branches are in- ␣ϭA•a, ͑A4͒ dependent of each other. with its inverse aϭAϪ1 ␣. Here A is a matrix given by • In the polariton representation, the Hamiltonian HЈpolariton w x y z describes two independent harmonic oscillators. From the 1 1 1 1 Heisenberg equation w2 x2 y 2 z2 Aϭ . ͑A5͒ ˆ y* z* w* x* dO 1 1 1 1 iប ϭ͓Oˆ ,H͔, ͑A9͒ dt ͩ y 2* z2* w2* x2*ͪ we obtain In the polariton representation, the Hamiltonian has the diagonal form ϪiE͑1͒t/ប ␣1k͑t͒ϭ␣1k͑0͒e k , ͑A10a͒

͑1͒ † 1 ͑2͒ † 1 ͑2͒ H ϭ E ϩ ϩE ϩ . ϪiEk t/ប polaritonЈ ͚ k ␣1k␣1k k ␣2k␣2k ␣2k͑t͒ϭ␣2k͑0͒e , ͑A10b͒ k ͫ ͩ 2ͪ ͩ 2ͪͬ ͑A6͒ or in a more compact form The subindices iϭ1 ,2 specify the two polariton branches, (1) (2) ␣͑t͒ϭU␣͑t͒␣͑0͒. ͑A11͒ with different dispersion relations Ek and Ek . The transformation matrix elements wi ,xi ,y i , and zi are determined Recall that the matrix form of the canonical transformation by requiring that the ␣ik’s satisfy boson commutation rela- from the photon and phonon operators (ak and bk) to the tions polariton operators (␣k)is␣ϭA•a. Thus at time t the photon and phonon operators can be expressed as , † ϭ , , ϭ0, A7 ͓␣ik ␣ jkЈ͔ ␦ij␦kkЈ ͓␣ik ␣ jkЈ͔ ͑ ͒ a͑t͒ϭAϪ1␣͑t͒ϭAϪ1U ͑t͒Aa͑0͒, ͑A12͒ so that ␣ which provides the time evolution of the photon and phonon ͑i͒ ͓␣ik ,H͔ϭEk ␣ik , ͑A8͒ operators.

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