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 1995! 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 period!, 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 p5\q and branch sub- tion is adequate. However, at low enough temperatures, script l has energy eql5\vql ; the corresponding creation quantum fluctuations become dominant. For example, a re- and annihilation operators satisfy the boson commutation re- cent study4 shows that quantum fluctuations in the atomic lations positions can indeed influence observable quantities ~e.g., the † Raman line shape! even when temperatures are not very low. @bq8l8,bql#5dqq8dll8, @bql ,bq8l8#50. ~1! With these facts in mind, and prompted by the many exciting developments in classical phonon optics, coherent phonon The atomic displacements uia of a crystal lattice are given by experiments, and ~on the other hand! squeezed states of 5 light, we would like to explore phonon analogs of quantum 1 N l l iq•Ri optics. In particular, we study the dynamical and quantum uia5 ( UqaQqe . ~2! fluctuation properties of the atomic displacements, in anal- ANm ql ogy with the modulation of quantum noise in light. Specifi- Here Ri refers to the equilibrium lattice positions, a to a cally, we study single-mode and two-mode phonon coherent l 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 l \ † Qq5A ~bql1b2ql!. ~3! proposed in the context of quantum optics. A coherent state 2vql 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: 0163-1829/96/53~5!/2419~6!/$06.0053 2419 © 1996 The American Physical Society 2420 XUEDONG HU AND FRANCO NORI 53 \ noiseless as the vacuum state. Coherent states are also the set Re@u ~q!#5 $Ul ~b 1b† ! of quantum states that best describe the classical harmonic a ( A8mv qa ql 2ql l ql oscillators.7 l † A single-mode phonon coherent state can be generated by 1U *~b2q 1b !%. ~4! qa l ql the Hamiltonian For simplicity, hereafter we will drop the branch subscript l, assume that U is real, and define a q-mode dimension- 1 qa H5\v b†b 1 1l*~t!b 1l ~t!b† ~14! less lattice amplitude operator: qS q q 2D q q q q † † u~6q!5bq1b2q1b2q1bq . ~5! and an appropriate initial state. Here lq(t) represents the interaction strength between phonons and the external This operator contains essential information on the lattice source. More specifically, if the initial state is a vacuum dynamics, including quantum fluctuations. It is the phonon state, uc(0)&5u0&, then the state vector becomes a single- analog of the electric field in the photon case. mode coherent state thereafter, Let us first consider the phonon vacuum state. When no 2ivqt phonon is excited, the crystal lattice is in the phonon vacuum uc~t!&5uLq~t!e &, ~15! state u0&. The expectation values of the atomic displacement and the lattice amplitude are zero, but the fluctuations will be where finite: i t L t 52 l t eivqtdt ~16! 2 2 2 q~ ! E q~ ! ^~Duia! &vac[^~uia! &vac2^uia& vac ~6! \ 2` N 2 is the coherent amplitude of mode q. If the initial state is a \uUqau 5( , ~7! single-mode coherent state uc(0)&5uaq&, then the state vec- q 2Nmvqa tor at time t takes the form 2 2ivqt ^@Du~6q!# &vac52. ~8! uc~t!&5u$Lq~t!1aq%e &, ~17! Let us now consider the phonon number states. The eigen- which is still a coherent state. 2ivqt states of the harmonic phonon Hamiltonian are number states In a single-mode (q) coherent state uLq(t)e &, ^uia(t)& coh and ^u(6q)&coh are sinusoidal functions of time. which satisfy bqunq&5Anqunq21&. 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 \uU u2 the phonon number is certain, as is the case with any number 2 qa ^~Duia! &coh5( . ~18! state unq&. Thus, in a number state, the expectation values of q 2Nmvqa the atomic displacement n u n and q-mode lattice am- ^ qu iau q& The unexcited modes are in the vacuum state and thus all plitude n u(6q) n vanish due to the randomness in the ^ qu u q& contribute to the noise in the form of zero-point fluctuations. phase of the atomic displacements. The fluctuations in a Furthermore, number state unq& are ^@Du~6q!#2& 52. ~19! U 2n N U 2 coh 2 \u qau q \u q8au ^~Du ! & 5 1 , ~9! 2 ia num Nmv ( 2Nmv From the expressions of the noise ^(Duia) &coh and qa q8Þq q8a 2 ^@Du(6q)# &coh , it is impossible to know which state ~if ^@Du~6q!#2& 5212n . ~10! any! has been excited, while this information is clearly num q present in the expression of the expectation value of the lat- tice amplitude ^u(6q)&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 bqubq&5bqubq&. ~11! 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(bq) to the phonon vacuum state coherent states. Here ‘‘quadrature’’ refers to the dimension- less coordinate and momentum. Compared to coherent states, † * ubq&5Dq~bq!u0&5exp~bqbq2bq bq!u0& ~12! squeezed ones can achieve smaller variances for one of the quadratures during certain time intervals and are therefore 2 ` nq ubqu bq helpful for decreasing quantum noise. Figures 1 and 2 sche- 5exp 2 ( unq&. ~13! matically illustrate several types of phonon states, including S 2 D n 50 q An ! 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 expec- tation value and the fluctuation of the lattice amplitude operator u(6q) in different states.