IL NUOVO CIMENTOVOL. 112 B, N. 6 Giugno 1997 NOTE BREVI
Classical analogue of a class of squeezed states
1 2 H. IOANNIDOU ( ) and D. SKALTSAS ( ) (1) Division of Applied Analysis, Department of Mathematics University of Patras - Patras, Greece (2) Department of Physics, University of Patras - Patras 26110, Greece
(ricevuto il 15 Luglio 1996; approvato il 25 Marzo 1997)
Summary. — The classical squeeze potential is a kinetic potential which generates an external force tending to remove a physical system from the state of equilibrium. The results of the classical model are compatible with those of quantum mechanics. PACS 03.65 – Quantum mechanics. PACS 02.90 – Other topics in mathematical methods in physics. PACS 42.50 – Quantum optics.
It has been shown that the unitary squeeze operator
(1) S(z) 4exp [ (za 12 2z * a 2 )O2] generates a potential of the form g(qp×1p×q)O2, which is added to the Hamiltonian operator of the harmonic oscillator, so that the prototype operator for the squeezed states derived by (1) is the quantum Hamiltonian [1, 2]
2 × ¯ ¯ ¯ (2) H 42ˇ2 N2m1mv2 q2 O22iˇg q 2 q N2. ¯q2 g ¯q ¯q h
The law of correspondence leads us to consider as the classical analogue of the operator (2), the classical Hamiltonian function
(3) H(q, P) 4P 2 O2m1mv 2 q 2 O21gqP . We note that the classical squeeze potential has the form U(q, P) 4qP. From the classical squeeze Hamiltonian we get the canonical system
dq dP (4) 4POm1gq , 42mv2 q2gP dt dt which coincides exactly with the quantum system resulting from the Heisenberg equations of motion [2]; accordingly, the solution of the system (4) is the same as it has
933 934 H. IOANNIDOU and D. SKALTSAS been derived in ref. [2]: i.e.
. 4 g 1 P0 1 q(t) q0 sin Vt q0 cos Vt , ` g V mV h (5) / 2 4 g 1 mv 1 ` P(t) P0 q0 sin Vt P0 cos Vt , ´ g V V h where V4kv 2 2g 2 , vDg. So far we have seen that both descriptions (classical and quantum) are equivalent in what concerns the dynamics of a squeezed system, and that neither of these offers more information. The Lagrangian corresponding to the Hamiltonian (3) found by the relation
dS 4P 2 O2m2mv 2 q 2 O2. dt is written in the standard form, as
dS(q, P) (6) 4 L(q, v) 4mv 2 O22m(v 2 2g 2 ) q 2 O22mgqv , dt where v4dqOdt4POm1gq. This Lagrangian involves a generalized potential energy V4m(v 2 2g 2) q 2 O21mgqv. The external force F can be expressed in the form [3]
¯V d ¯V (7) F42 1 42(v22g2) q . ¯q dt g ¯v h
This force is composed by two components, one tending to return the system to the centre of oscillation, and the other tending to remove the system. The squeeze potential mgqv is the kinetic potential obeying the equation
d ¯ mgqv40. g dt ¯v h
The Hamilton-Jacobi equation with Hamiltonian (2), is
1 ¯S 2 1 ¯S ¯S (8) 1 mv 2 q 2 1gq 1 40. 2m g ¯q h 2 ¯q ¯t
In solving the above equation by use of the method of the characteristic strips [4], we get the following characteristic solutions:
(9) S (1)(q)4m(iV2g) q2 O2,
(10) S (2)(q)42m(iV1g) q2 O2, CLASSICAL ANALOGUE OF A CLASS OF SQUEEZED STATES 935 where
V4kv 2 2g 2 D0, which are complex and conjugate to each other. The function S (1) apperars identically in the exponential of the “squeezed” eigenfunctions derived by Jannussis and Skaltsas in ref. [2]. These eigenfunctions for vDg are
4 O 1O4 n 21O2 2 1 2 O O (11) c n (q) (mV pˇ) (n!2 ) exp [ (mV img) q 2ˇ] Hn (kmV ˇQq) and agree with the de Broglie form of the wave function
4 iS (1)(q)Oˇ c n yn (q) e , where S (1) is our function equation (9). The second solution S (2) is not acceptable, because of the factor iS (2)4(mV2img) q 2 O2, leading to states of the form exp [mVq 2 O2ˇ], which are not normalizeable for VD0. The probability densities of the eigenstates result from eq. (11) as
1O2 (mVOpˇ) 2 4 4 Oˇ 2mVq Oˇ (12) r n (q) c n c*n Hn(kmV Qq)e , n!2n and we distinguish the factor of the normal distribution. The classical probability density can be derived from the equation of continuity ¯v ¯r ¯r (13) r 1v 1 40. ¯q ¯q ¯t 1 ˇS In this equation we have to consider v4v(q, t) 4 1gq. By means of our solutions (9) and (10), eq. (13) becomes m ˇq ¯r (14) V r1q 40. g ¯q h Having in mind that the function r must be such that Q r dq41, 2Q we see that eq. (14) is satisfied by the distribution 4 2 O 4 (15) r(q) r 0 d[ (q q0 ) a]; r0, a const , where d is the Dirac delta-function. This function is the limit of a normal distribution of the form
2 2 2 O A (q q0 ) B r r 0 e for vanishing variance. We conclude that the classical probability density eq. (15) is the limit of the corresponding quantum one for ˇK0. In the classical case the “squeezing” is stronger, since the particle is forced to remain in the initial state of equilibrium. Also the uncertainty of the position vanishes, as it is expected for a classical particle. On the 936 H. IOANNIDOU and D. SKALTSAS whole we can say that classical squeezed states supply the same results with the quantum approach in what concerns the dynamical behavior, but stronger squeezing in what concerns distributions, uncertainties and mean values. the classical study permits us to see the cause of squeezing, which is the application of an external force 2g 2 q, to photons with frequency v. The presented study refers to the class of squeezed states described by the operator equation (1). There are indications [5] that different unitary operators might lead to different results.
*** The authors wish to thank Prof. A. D. JANNUSSIS for useful discussions.
R E F E R E N C E S
[1] JANNUSSIS A. and BARTZIS V., Phys. Lett. A, 132 (1988) 324. [2] JANNUSSIS A. and SKALTSAS D., Nuovo Cimento B, 104 (1989) 17. [3] WHITTAKER E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge University Press) 1965. [4] COURANT R. and HILBERT D., Methods in Mathematical Physics, Vol. II (Interscience, New York) 1962. [5] DE OLIVEIRA F. A., PEDROSA I. A. and BASEIA B., Nuovo Cimento B, 108 (1993) 275.