Volume 148, number 1,2 PHYSICS LETTERS A 6 August 1990 Quantum non-demolition stroboscopic observables and multipumping back-action evasion schemes Roberto Onofrio Scuola delDottorato di Ricerca, Dipartimento di Fisica, Università di Roma ‘La Sapienza”, Rome, Italy and INFN, Sezione di Roma, Rome, Italy Received 11 April 1989; revised manuscript received 9 May 1990; accepted for publication 6 June 1990 Communicated by J.P. Vigier It is shown that quantum non-demolition stroboscopic measurements of a complex amplitude of a harmonic oscillator are obtained as a limit of a multipump back-action evasion scheme. As a consequence of the analogy between these two strategies a new stroboscopic scheme forthe detection of small displacements in macroscopic mechanical oscillators is proposed. Quantum mechanics seems to introduce relevant — limitations on the sensitivity for the measurements F 1~ a] — of small displacements in a macroscopic mechanical This is satisfied if 0, depends only on 0a and the oscillator [1]. A wide range of rneasunng apparatus other simultaneously diagonalizable observables ?,~ may be schematized by means of a set of macro- ([ia, i~] = 0). A particular example is that when scopic mechanical oscillators, especially those de- the interaction Hamiltonian depends linearly on veloped for detecting gravitational waves and, more through a function O( ~p) of the observables 2~ as in general, for experimental gravitation researches. 2 Thus, the quantum mechanical limitations seem to rii=O(Yp) Aa ( ) give ultimate limits on the possibility to test in lab- and will be of relevance in the following con- oratory post-Newtonian effects such as those pre- siderations. dicted by metric theories of gravitation, general rel- Furthermore the repeatibility of the measurement ativity in particular [2]. It was recognized, however, of 5~ain a non-demolition way, i.e. without Heisen- that non-relativistic quantum mechanics allows berg uncertainties in this observable during the mea- strategies, called quantum non-demolition (QND), surernent time set ‘r: t= {t 0, t1, ..., t,~}, is ensured if~a in which a repeated set of high accuracy measure- commutes with itself in the set ‘t, i.e. ments of only one observable is possible [3—5].The Heisenberg uncertainty, always present in a mea- [~‘a(tj), ~a(tj)] 0 Vt1, t1et. (3) surernent, will be, of course, enhanced in the other If ‘r is a discrete set the strategy is also called QND non-commuting observables. According to the quan- stroboscopic, as opposed to the other situation of a turn theory of measurement two conditions have to continuous set corresponding to a so-called QND be satisfied in order to perform a QND strategy continuous strategy. [6,7]. The conditions (2) and (3) may be applied for In the first place an unperturbed measurement of the search of QND strategies on an electromechan- the observable ~a is obtained if the interaction ical transducer. In a general schematization of an Hamiltonian operator 0, (i.e. the term due to the electromechanical transducer we can consider two interaction between the observed system and the oscillators, one mechanical and one electrical with measuring apparatus) commutes with ~ coordinates and momenta (.~, ft) and (~, ft) respec- 0375-9601/90/S 03.50 © 1990 — Elsevier Science Publishers B.V. (North-Holland)