Quantum jump dynamics in cavity QED - Institut Fourier

JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 43, NUMBER 7

JULY 2002

Quantum jump dynamics in cavity QED D. Spehnera) and M. Orszag Facultad de Fı´sica, Pontificia Universidad Cato´lica de Chile, Casilla 306, Santiago 22, Chile

共Received 18 December 2001; accepted for publication 25 February 2002兲 We study the stochastic dynamics of the electromagnetic field in a lossless cavity interacting with a beam of two-level atoms, given that the atomic states are measured after they have crossed the cavity. The atoms first interact at the exit of the cavity with a classical laser field E and then enter into a detector which measures their states. Each measurement disentangles the field and the atoms and changes in a random way the state 兩 ␺ (t) 典 of the cavity field. For weak atom-field coupling, the evolution of 兩 ␺ (t) 典 when many atoms cross the cavity and the detector is characterized by a succession of quantum jumps occurring at random times, separated by quasi-Hamiltonian evolutions, both of which depend on the laser field E. For E⫽0, the dynamics is the same as in the Monte Carlo wave function model of Dalibard et al. 关Phys. Rev. Lett. 68, 580 共1992兲兴 and Carmichael, An Open System Approach to Quantum Optics, Lecture Notes in Physics Vol. 18 共Springer, Berlin, 1991兲兴. The density matrix of the quantum field, obtained by averaging the projector 兩 ␺ (t) 典具 ␺ (t) 兩 over all results of the measurements, is independent of E and follows the master equation of the damped harmonic oscillator at finite temperature. We provide numerical evidence showing that for large E, an arbitrary initial field state 兩␺共0兲典 evolves under the monitoring of the atoms and the measurements toward squeezed states 兩 ␣ ,re 2i ␾ 典 , moving in the ␣-complex plane but with almost constant squeezing parameters r and ␾. The values of r and ␾ are determined analytically. On the other hand, for E⫽0, the dynamics transforms the initial state into Fock states 兩 n 典 with fluctuating numbers of photons n, as shown in Kist et al. 关J. Opt. B: Quantum Semiclassical Opt. 1, 251 共1999兲兴. In the last part, we derive the quantum jump dynamics from the linear quantum jump model proposed in Spehner and Bellissard 关J. Stat. Phys. 104, 525 共2001兲兴, for arbitrary open quantum systems having a Lindblad-type evolution. A careful derivation of the infinite jump rates limit, where the dynamics can be approximated by a diffusion process of the quantum state, is also presented. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1476392兴

I. INTRODUCTION

The dissipative dynamics of an open quantum system S can be described in two different ways. The first one consists in coupling S with a reservoir R and assuming that the total system S⫹R is isolated. Since one is concerned by the dynamics of S only, one traces out the degrees of freedom of R in the equation of motion of S⫹R. Within the Markov approximation, the reduced density matrix of S follows a first-order differential equation with time-independent coefficients. In many cases, a separation of time scales between the Hamiltonian 共R-independent兲 and dissipative 共R-dependent兲 evolutions allows one to perform a local averaging in time, which kills nonresonant terms.1,2 The coarse-grained master equation obtained in this way has the Lindblad form.3 An alternative approach to this density matrix description has been developed in the last two a兲

Present address: Universita¨t Essen, Fachbereich Physik, D-45117 Essen, Germany; electronic mail: [email protected]

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© 2002 American Institute of Physics

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J. Math. Phys., Vol. 43, No. 7, July 2002

D. Spehner and M. Orszag

decades in quantum optics4 – 8 共see, e.g., Ref. 9兲, quantum measurement theory,10–13 quantum and classical stochastic calculus,14 –17 and electronic transport in solids.18 –20 This approach is based on stochastic evolutions of pure states. The system is described by a random wave function 共RW兲 evolving according to a linear or nonlinear stochastic Schro¨dinger equation. Consistency with the density matrix approach requires that the pure state evolution gives the master equation back after averaging over the dynamical noise. Apart from being intuitively appealing, the RW models provide quite efficient tools for solving master equations numerically, since Schro¨dinger equations have N components, whereas master equations have N⫻N components, N being the dimension of the Hilbert space of S. However, the RW models are more than simple mathematical or numerical tools: they describe the real evolution of single quantum systems under continuous monitoring by measurements 共photon counting, homodyne or heterodyne detections兲.5,8,9 In recent years, the attainment of low temperature and low dissipation regimes, as well as the improvements of detection techniques, has allowed the investigation of the dynamics of such continuously monitored systems. Remarkable examples of these are single ions21 and Bose–Einstein condensates22 in electromagnetic traps, probed by laser beams, and electromagnetic fields in high Q-cavities,23 probed by beams of highly excited atoms 共Rydberg atoms兲. This new generation of experiments, combined with the difficulties usually encountered in solving the master equation, has strongly stimulated the developments of the RW approach in quantum optics. The aim of this paper is to investigate a specific physical realization, which could be in principle realizable experimentally 共although this question is not addressed here兲, of a class of RW models based on quantum jumps. The system we consider is the electromagnetic field of a high Q-cavity interacting with a beam of two-level atoms, which forms the reservoir of temperature T. The states of the atoms leaving the cavity are measured by a detector. A laser field E is placed between the cavity and the detector. The corresponding master equation, obtained by averaging over the results of the measurement on the atoms, is, for weak atom-field coupling, the equation of the damped harmonic oscillator with finite temperature. The same problem has been considered in Ref. 24 in the reverse situation where one knows exactly the state of each atom before it crosses the cavity and no measurement is performed on it at the exit 共its final state thus being unknown兲. It has been shown in this reference that the cavity field evolves at large times to a state which is completely controlled by the atomic initial states. We first introduce in Sec. II the class of quantum jump models studied in this work, for arbitrary open systems having a Lindblad-type evolution. The experimental scheme is presented in Sec. III, where we also compute the random evolution of the cavity field and its corresponding average evolution. We focus in Sec. IV on single quantum trajectories, i.e., single realizations of the measurements. The numerical simulations and analytical results presented in this section show that for T⬎0 and large laser fields E, the state of the cavity field localizes at large times to squeezed states with an almost constant squeezing amplitude r which depends only on T. Section V is devoted to the derivation, for arbitrary open systems, of the nonlinear quantum jump schemes from the corresponding linear ones introduced in Ref. 18. Their relation with the so-called quantum state diffusion stochastic Schro¨dinger equations10–12,15–17 is also established. Our conclusions are presented in Sec. VI. II. THE QUANTUM JUMP SCHEMES

Let us first recall briefly a few basic facts about the master equation approach to open quantum systems. Consider an open system S interacting with a reservoir R. The density matrix ␴ of the total system S⫹R is assumed to follow the Liouville–von Neumann equation of closed systems. A state of S is specified by the reduced density matrix ␳, defined as the partial trace of ␴ over the reservoir’s Hilbert space. By tracing out the degrees of freedom of R in the Liouville–von Neumann equation, one obtains an integro-differential equation for ␳ 共Nakajima–Zwanzig equation兲.25 Using a Born–Markov approximation and a local time averaging on a time scale much larger than the inverse Bohr frequencies of S, this equation is transformed into a simpler first-order linear differential equation, called the master equation.1,25 This coarse-grained equation has in most cases the Lindblad form:3



Quantum jump dynamics in cavity QED

d␳ 1 ⫽⫺i关 H, ␳ 兴 ⫹ dt 2

兺m 共关 L m ␳ ,L m† 兴 ⫹ 关 L m , ␳ L m† 兴兲 .

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共1兲

H is the Hamiltonian of S 共including the energy shifts due to the coupling with the reservoir兲, and L m are some operators acting on the Hilbert space of S. The sum over the discrete indices m can be finite or infinite, depending on the nature of the problem. We now describe the random wave function approach of Dalibard, Castin, and Mølmer4 and Carmichael.5 This approach, called in the former reference the Monte Carlo wave function method, has been introduced independently by several other authors.6,7,13 It is based on quantum jumps, i.e., on discontinuous random evolutions of the wave function of S. At some random times, quantum jumps 共QJ兲 occur as a result of some continuous measurement on the system S 共e.g., a detection of a photon emitted by a system constituted by an atom兲. If a jump occurs, the wave function 兩␺典 of S is modified discontinuously as follows: jump m:

兩␺典→

L m兩 ␺ 典 , 储 L m兩 ␺ 典储

共2兲

where L m are the Lindblad operators appearing in 共1兲. The probability of occurrence of a jump of type m in the time interval 关 t,t⫹ ␦ t 兴 is

␦ p m 共 t 兲 ⫽ 储 L m 兩 ␺ 共 t 兲典储 2 ␦ t.

共3兲

One must choose ␦ t small enough so that ␦ p m (t)Ⰶ1 for any 兩␺典 and all m’s 共this is fulfilled if ␦ t ⫺1 is much bigger than the damping constants ␥ m appearing in the master equation, contained in the operators L m 兲. If no jump occurs between t and t⫹ ␦ t, the wave function evolves between these two times according to Schro¨dinger’s equation with an effective Hamiltonian H⫹K, and is then normalized at t⫹ ␦ t: 兩 ␺ 共 t⫹ ␦ t 兲典 ⫽

兩 ␸ 共 t⫹ ␦ t 兲典 , 储 ␸ 共 t⫹ ␦ t 兲储

兩 ␸ 共 t⫹ ␦ t 兲典 ⫽e ⫺i ␦ t(H⫹K) 兩 ␺ 共 t 兲典 .

共4兲

K can be computed in special cases by first determining perturbatively the wave function of the total system S⫹R and then projecting it onto the subspace corresponding to the no-jump measurement.4,9 An easier 共though less fundamental兲 way to compute K is to ask directly that the average M兩 ␺ (t) 典具 ␺ (t) 兩 satisfies the master equation 共1兲共see the following兲. This gives4 K⫽K 0 ⬅

1 2i

兺m L m† L m .

共5兲

Note that K 0 is not self-adjoint. Hence the norm of the wave function is not conserved by the evolution operator e ⫺i ␦ t(H⫹K 0 ) . This can be interpreted by invoking the gain of information on the system provided by the measurement, namely, by the knowledge that no jump occurred between t and t⫹ ␦ t. For instance, in the case of an atom coupled to the quantized electromagnetic field, we may infer from a no-photon detection that the atom has not emitted spontaneously a photon. Since the wave function is normalized at each step ␦ t in 共2兲 and 共4兲, the random dynamics is norm-preserving on the time resolution ␦ t. These normalizations make the stochastic quantum evolution nonlinear. We will see in Sec. V that it is possible to define an equivalent linear model, in which the random wave function is not normalized.18,14 The map t苸关 0,⬁ 关哫兩 ␺ (t) 典 for a given outcome of the jumps is called a quantum trajectory.5 Let us consider the average density matrix ␳ (t)⫽M兩 ␺ (t) 典具 ␺ (t) 兩 , where M is the average over all realizations of the jumps. It can be easily shown4 that ␳ (t) obeys the master equation 共1兲 to lowest order in 储 L m 储 2 ␦ t. Actually, taking H⫽0 for simplicity, one has


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␳ 共 t⫹ ␦ t 兲 ⫽Mt⫹ ␦ t 兩 ␺ 共 t⫹ ␦ t 兲典具 ␺ 共 t⫹ ␦ t 兲兩 ⫽Mt

冉冉

1⫺

兺m

冊

†

e ⫺i ␦ tK 0 兩 ␺ 共 t 兲典具 ␺ 共 t 兲兩 e i ␦ tK 0 ␦ p m共 t 兲 ⫹ 储 e ⫺i ␦ tK 0 兩 ␺ 共 t 兲典储 2

兺m

␦ p m共 t 兲

冊

† L m兩 ␺ 共 t 兲具 ␺ 共 t 兲兩 L m . 储 L m兩 ␺ 共 t 兲典储 2

This equation is simplified by taking

冉

e ⫺i ␦ tK 0 ⫽ 1⫺ ␦ t

兺m

† Lm Lm

冊

1/2

共6兲

.

Then, by 共3兲:

冉

␳ 共 t⫹ ␦ t 兲 ⫽ 1⫺ ␦ t

兺m L m† L m

冊冉 1/2

␳ 共 t 兲 1⫺ ␦ t

兺m L m† L m

冊

1/2

⫹␦t

兺m L m ␳ 共 t 兲 L m† .

共7兲

Expanding the square roots and keeping only terms of order one in ␦ t, one obtains 共1兲 and K 0 is given by 共5兲. Note that, for an arbitrary time interval ␦ t between consecutive measurements, ␳ (t⫹ ␦ t) is not given by integrating 共1兲 from t to t⫹ ␦ t 共a different result is already obtained at the next order ␦ t 2 兲. This should be kept in mind when dealing with real or numerical experiments, where ␦ t is always finite. ⬘ on the operators L m which does not change 共1兲. The Consider a transformation L m →L m quantum jumps and the effective Hamiltonian K may be modified by this replacement. This leads to a different stochastic dynamics, which unravels the same master equation. A particular transformation leaving 共1兲 invariant is26

⬘ ⫽L m ⫹␭ m , L m →L m

H→H ⬘ ⫽H⫹

1 2i

兺m 共 ␭ m* L m ⫺␭ m L m† 兲 ⫽H ⬘ † ,

共8兲

where ␭ m ’s are complex numbers. This invariance of the Lindblad equation is not related to a particular symmetry of the system or its coupling with the reservoir. It simply expresses that the separation between the Hamiltonian part ⫺i 关 H, ␳ 兴 and the remaining dissipative part in 共1兲 is not unique. The transformation 共8兲 generates a whole family of distinct QJ models depending on the set of numbers ␭ m . The modification of the wave function at a jump m is now given by jump m:

兩␺典→

W m兩 ␺ 典储 W m兩 ␺ 典储

共9兲

with the jump operators W m proportional to (L m ⫹␭ m ): ⫺1/2 W m⫽ ␥ m 共 L m ⫹␭ m 兲 .

共10兲

⬘ in 共5兲 and adding to The new generalized Hamiltonian H⫹K is obtained by replacing L m by L m it H ⬘ ⫺H: K⫽

1 2i

兺m 共 L m† L m ⫹2␭ m* L m ⫹ 兩 ␭ m兩 2 兲 .

共11兲

The last term in the sum, proportional to the identity operator, is written only for convenience: it is irrelevant because of the normalization in Eq. 共4兲 giving the evolution between jumps. The probability of occurrence of a jump of type m becomes

␦ p m 共 t 兲 ⫽ 储共 L m ⫹␭ m 兲兩 ␺ 共 t 兲典储 2 ␦ t⫽ ␥ m ␦ t 储 W m 兩 ␺ 共 t 兲典储 2 .

共12兲

Note that it increases like 兩 ␭ m 兩 2 for large ␭ m .




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FIG. 1. The two-level atoms of the beam cross one by one a cavity containing the studied quantum field a, a second cavity containing a laser field E, and a detector measuring their states.

III. FIELD MODE IN A CAVITY

It is shown in this section that the QJ schemes described in Sec. II can be physically realized by an atomic beam crossing a cavity with perfectly reflecting walls and interacting with the quantum field inside, which forms the system S. The measurements are performed on the outgoing atoms, after they have interacted with a classical laser field E placed between the cavity and the measuring apparatus. A. Experimental scheme

Let us consider one mode of the quantized electromagnetic field of a lossless cavity coupled to its environment. The environment is a beam of atoms prepared in one of two Rydberg states 兩 g 典共‘‘ground state’’兲 and 兩 e 典共‘‘excited state’’兲 in resonance with the frequency ␻ of the mode. The fluxes r g and r e of atoms crossing the cavity, prepared, respectively, in states 兩 g 典 and 兩 e 典 , are assumed to be such that at most one atom is in the cavity at any time. The time interval between the crossing of two consecutive atoms in the cavity is ␦ t⫽(r g ⫹r e ) ⫺1 . To simplify, all the atoms of the beam are supposed to have the same speed. They thus spend the same time ␶ ⬍ ␦ t in the cavity, interacting with the field mode. The atom-mode interaction Hamiltonian is in the interaction picture 共rotating-wave approximation兲:1,2 H int⫽⫺i 共 ␭ * 兩 g 典具 e 兩 a † ⫺␭ 兩 e 典具 g 兩 a 兲 ,

共13兲

where a † and a are the creation and annihilation operators of a photon. The coupling constant ␭ ជ the is equal to 冑q 2 ␻ /2␧ 0 V ជd ge "␴ជ , where dជ ge is the matrix element of the atomic dipole, ␴ polarization vector of the field mode, q the charge of the electron, and V the volume of the cavity. At the exit of the cavity, the atoms enter into a second cavity, identical to the first but containing a classical laser field Eជ 共Fig. 1兲. They spend a time ␶ L ⬍ ␦ t there, interacting with the laser field. Under the dipolar and rotating-wave approximations, the atom-laser interaction Hamiltonian is in the interaction picture:1,2 i H L ⫽⫺ 共 ⍀ * 兩 g 典具 e 兩 ⫺⍀ 兩 e 典具 g 兩兲 , 2

共14兲

where ⍀⫽i dជ ge "Eជ is the Rabi frequency. The Hamiltonian 共14兲 describes the atom-laser interaction for a laser field in resonance with the atomic transition. The more general situation of nonzero detuning of the laser frequency will be discussed in the following. Finally, the state of each atom at the exit of the second cavity is measured by a detector, telling us if it is in its ‘‘ground’’ or in its ‘‘excited’’ state. The corresponding experimental scheme is presented in Fig. 1. It has been considered in Ref. 27 without the laser field E. The flight times of the atoms between the two cavities and between the second cavity and the detector are taken sufficiently small so that spontaneous emission of photons by the atoms can be neglected.


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B. Stochastic dynamics of the field mode

Let us compute the evolution of the state of the field mode in the first cavity, for a given result of the measurements. If one looks at it with a time resolution equal or bigger than the time ␦ t separating the entrance of consecutive atoms, the field mode is continuously monitored by the measurements on the atoms 共continuous measurement兲. The coupling constant ␭, the Rabi frequency ⍀, and the interaction times ␶ , ␶ L are assumed to satisfy 兩 ␭ 兩 ␶ Ⰶ1,

兩 ⍀ 兩 ␶ L Ⰶ1.

共15兲

Let us first determine the change of the wave function 兩 ␺ (t) 典 of the field mode when one atom, initially in state 兩 i 典 , i⫽g or e, crosses the two cavities and the detector. At the time t just prior to the entrance of the atom in the first cavity, the wave function 兩 ⌿(t) 典 of the total system ‘‘atom⫹field mode’’ is a tensor product state 兩 ⌿(t) 典 ⫽ 兩 i 典兩 ␺ (t) 典共the atom and the field did not yet interact兲. Since the two fields are in separated cavities, the atom interacts with the quantum field before interacting with the classical field E. The total wave function at the exit of the second cavity 共before the measurement兲 is thus, in the interaction picture, 兩 ⌿ ent典 ⫽e ⫺i ␶ L H L e ⫺i ␶ H int兩 i 典兩 ␺ 共 t 兲典 .

共16兲

Note that the evolution would be different if the laser field was placed in the first cavity 共in this case, the product of exponentials must be replaced by the exponential of H int⫹H L ). Because of their interaction, the quantum field and the atom are now entangled, i.e., 兩 ⌿ ent典 is not a product state. After the measurement on the atom has been performed, the field and the atom again become disentangled and the wave function of the total system is 兩 ⌿ 共 t⫹ ␦ t 兲典 ⫽ 兩 j 典兩 ␺ 共 t⫹ ␦ t 兲典 ,

兩 ␺ 共 t⫹ ␦ t 兲典 ⫽

具 j 兩 ⌿ ent典 , 储具 j 兩 ⌿ ent典储

共17兲

with j⫽g if the atom is detected in its ground state and j⫽e if it is detected in its excited state. We have assumed for simplicity that the measurement is performed at a time t⫹ ␶ mes⬍t⫹ ␦ t earlier than the entrance of the next atom in the first cavity. Since we work in the interaction picture, 兩 ⌿(t⫹ ␶ mes) 典 is then equal to 兩 ⌿(t⫹ ␦ t) 典 . The wave function 兩 ␺ (t⫹ ␦ t) 典 of the mode at the time t⫹ ␦ t is thus well-defined and given by 兩 ␺ 共 t⫹ ␦ t 兲典 ⫽

兩 ␸ 共 t⫹ ␦ t 兲典 , 储 ␸ 共 t⫹ ␦ t 兲储

兩 ␸ 共 t⫹ ␦ t 兲典 ⫽ 具 j 兩 e ⫺i ␶ L H L e ⫺i ␶ H int兩 i 典兩 ␺ 共 t 兲典 .

共18兲

The probability that the atom is detected in state 兩 j 典 , given that it enters in the cavity in state 兩 i 典 , is p i→ j ⫽ 储具 j 兩 ⌿ ent典储 2 ⫽ 储 ␸ 共 t⫹ ␦ t 兲储 2 .

共19兲

A straightforward computation leads to ⬁

e

⫺i ␶ H int

⫽

兺 p⫽0 ⬁

⫹

共 ⫺兩␭兩2␶ 2 兲 p 共兩 g 典具 g 兩 n p ⫹ 兩 e 典具 e 兩共 n⫹1 兲 p 兲共 2p 兲 !

兺 p⫽0

共 ⫺兩␭兩2␶ 2 兲 p 共 ␭ ␶ 兩 e 典具 g 兩 a n p ⫺␭ * ␶ 兩 g 典具 e 兩 n p a † 兲 , 共 2p⫹1 兲 !

where n⫽a † a is the number operator. Denoting by n 1/2 its square root, this formula can be rewritten in a more compact form:




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e ⫺i ␶ H int⫽ 兩 g 典具 g 兩 cos共兩 ␭ 兩 ␶ n 1/2兲 ⫹ 兩 e 典具 e 兩 cos共兩 ␭ 兩 ␶ 共 n⫹1 兲 1/2兲 ⫹␭ ␶ 兩 e 典具 g 兩 a sinc共兩 ␭ 兩 ␶ n 1/2兲 ⫺␭ * ␶ 兩 g 典具 e 兩 sinc共兩 ␭ 兩 ␶ n 1/2兲 a †

共20兲

with sinc(u)⫽(sin u)/u. Similarly, e ⫺i ␶ H L ⫽cos

冉冊

冉冊

冉冊

兩⍀兩␶L 兩⍀兩␶L 兩⍀兩␶L ⍀␶L ⍀ *␶ L ⫹ 兩 e 典具 g 兩 sinc ⫺ 兩 g 典具 e 兩 sinc . 2 2 2 2 2

Let us introduce the operator ã ⫽a sinc共兩 ␭ 兩 ␶ n 1/2兲

共21兲

and the complex numbers

␩ ⫽␭ ␶ ,

⑀⫽

⍀ tan共兩 ⍀ 兩 ␶ L /2兲 . 兩⍀兩␭␶

共22兲

Four cases must be considered. 共1兲 The atom enters the first cavity in its ground state and is detected in the same state: i⫽ j⫽g. Then, 兩 ␸ 共 t⫹ ␦ t 兲典 ⫽ 共 1⫹ 兩 ␩ ⑀ 兩 2 兲 ⫺ 1/2共 cos共兩 ␩ 兩 n 1/2兲 ⫺ 兩 ␩ 兩 2 ⑀ * ã 兲兩 ␺ 共 t 兲典 ⬅ 共 1⫺i ␦ t K g 兲兩 ␺ 共 t 兲典 .

共23兲

共2兲 The atom enters the first cavity in its ground state and is detected in its excited state: i⫽g, j⫽e. Then, 兩 ␸ 共 t⫹ ␦ t 兲典 ⫽ ␩ 共 1⫹ 兩 ␩ ⑀ 兩 2 兲 ⫺ 1/2 W ⫺ 兩 ␺ 共 t 兲典 ,

W ⫺ ⬅a ˜ ⫹ ⑀ cos共兩 ␩ 兩 n 1/2兲 .

共24兲

共3兲 The atom enters the first cavity in its excited state and is detected in its ground state: i⫽e, j⫽g. Then, 兩 ␸ 共 t⫹ ␦ t 兲典 ⫽⫺ ␩ * 共 1⫹ 兩 ␩ ⑀ 兩 2 兲 ⫺ 1/2 W ⫹ 兩 ␺ 共 t 兲典 ,

W ⫹ ⬅a ˜ † ⫹ ⑀ * cos共兩 ␩ 兩共 n⫹1 兲 1/2兲 .

共25兲

共4兲 The atom enters the first cavity in its excited state and is detected in the same state: i⫽ j⫽e. Then, 兩 ␸ 共 t⫹ ␦ t 兲典 ⫽ 共 1⫹ 兩 ␩ ⑀ 兩 2 兲 ⫺ 1/2共 cos共兩 ␩ 兩共 n⫹1 兲 1/2兲 ⫺ 兩 ␩ 兩 2 ⑀ ã † 兲兩 ␺ 共 t 兲典 ⬅ 共 1⫺i ␦ t K e 兲兩 ␺ 共 t 兲典 .

共26兲

Cases 共2兲 and 共3兲 correspond, respectively, to the absorption and the emission of a photon of the cavity mode or of the laser field by the atom. In order to have quantum jumps separated by ‘‘continuous’’ Hamiltonian evolutions on time scales bigger than ␦ t, the probabilities of these events must be very small. The probability ␦ p ⫺ (t) of case 共2兲 is equal to p g→e multiplied by the probability r g ␦ t that the atom enters in the first cavity in state 兩 g 典 . With the help of 共19兲 and 共24兲, one gets

␦ p ⫺共 t 兲 ⫽

rg ␦t 兩␩兩2 储 W ⫺ 兩 ␺ 共 t 兲典储 2 ⫽r g ␦ t 共储 sin共兩 ␩ 兩 n 1/2兲兩 ␺ 共 t 兲典储 2 ⫹O共 ␩ ⑀ 兲 ). 1⫹ 兩 ␩ ⑀ 兩 2

共27兲

The probability ␦ p ⫹ (t) of case 共3兲 is given by a similar formula, replacing r g by r e , W ⫺ by W ⫹ , and n by n⫹1. The probabilities of cases 共1兲 and 共4兲 are, respectively, r g ␦ t⫺ ␦ p ⫺ (t) and r e ␦ t ⫺ ␦ p ⫹ (t). We see that ␦ p ⫾ (t) is small if 兩 ␩ 兩 ⫽ 兩 ␭ 兩 ␶ Ⰶ1 and 兩 ␩ ⑀ 兩 ⯝ 兩 ⍀ 兩 ␶ L /2Ⰶ1, provided that also

具 ␺ 共 t 兲兩共兩 ␩ 兩 n 1/2⫺k ␲ 兲 2q 兩 ␺ 共 t 兲典 ⫽O共 ␩ q 兲 ,

q⫽1,2, . . . ,

共28兲


3518



with k a non-negative integer. This condition is met for k⫽0 if the maximal number of photons in the cavity is much smaller than 兩 ␩ 兩 ⫺2 共of order 兩 ␩ 兩 ⫺1 or smaller兲. This corresponds to the perturbative regime. If the condition is met for k⭓1, the atom makes almost k/2 Rabi oscillations in the cavity and leaves it, with a high probability, nearly in the same state as it entered it. Indeed, neglecting terms of order ␩ 2 and ␩ 4 ⑀ 4 , it follows from 共22兲, 共23兲, and 共26兲:

␦ tK g ⯝

兩␩兩2 † 共 ã ã ⫹2 共 ⫺1 兲 k ⑀ *ã ⫹ 兩 ⑀ 兩 2 兲 , 2i

共29兲

兩␩兩2 ␦ tK e ⯝ 共 ã ã † ⫹2 共 ⫺1 兲 k ⑀ ã † ⫹ 兩 ⑀ 兩 2 兲 , 2i

up to an irrelevant additional phase k ␲ . When the atom is measured in the same state as its initial state 关cases 共1兲 and 共4兲兴, the normalized wave function of the mode is thus modified, up to a sign, by an amount of order 兩 ␩ 兩 ⫹ 兩 ␩ 3/2⑀ 兩 . In the opposite cases 共2兲 and 共3兲, the mode wave function is modified by an amount of order 1. It suffers a quantum jump: 兩 ␺ 共 t⫹ ␦ t 兲典 ⫽

W ⫾兩 ␺ 共 t 兲典 , 储 W ⫾兩 ␺ 共 t 兲典储

共30兲

with the jump operators ˜ ⫹ 共 ⫺1 兲 k ⑀ , W ⫺ ⯝a

共31兲

W ⫹ ⯝a ˜ † ⫹ 共 ⫺1 兲 k ⑀ * .

The signs ⫺ and ⫹ correspond to case 共2兲共absorption of a photon兲 and case 共3兲共emission of a photon兲, respectively. The probability that a jump ⫾ occurs is of order 兩 ␩ 兩 ⫹ 兩 ␩ 3/2⑀ 兩 ⫹ 兩 ␩ ⑀ 兩 2 , see 共27兲 and 共28兲. It is given approximately by

␦ p ⫾共 t 兲 ⯝ ␥ ⫾␦ t 储 W ⫾兩 ␺ 共 t 兲典储 2,

共32兲

where we have introduced the damping rates:

␥ ⫺ ⫽r g 兩 ␩ 兩 2 ⫽r g 兩 ␭ 兩 2 ␶ 2 ,

共33兲

␥ ⫹ ⫽r e 兩 ␩ 兩 2 ⫽r e 兩 ␭ 兩 2 ␶ 2 . The operator ã is given, up to terms of order ␩ 1/2, by

ã ⯝

再

a

if k⫽0

共 ⫺1 兲 k a

冉

兩 ␩ 兩 n 1/2⫺k ␲ 共兩 ␩ 兩 n 1/2⫺k ␲ 兲 2 ⫺ k␲ 共 k␲ 兲2

冊

if k⭓1.

共34兲

If 兩 ␺ (t) 典 ⫽ 兩 n 典 is a Fock state with n⫽(k ␲ / 兩 ␩ 兩 ) 2 photons, the crossing of the atom of initial state 兩 i 典 has no effect on the field if i⫽g 共since ã 兩 ␺ (t) 典 ⫽0), and a small effect if i⫽e and 兩 ␩ 兩 Ⰶ1. If condition 共28兲 is not met, the atom strongly modifies the state of the field mode in all cases 共1兲–共4兲. Hence there is no ‘‘continuous-like’’ Hamiltonian evolution changing weakly the state of the mode, separated by unlikely jumps. For ⑀ ⫽0, the evolutions for i⫽g 关Eq. 共23兲兴 and i⫽e 关Eq. 共26兲兴 have the form 共6兲共the jump operators W ⫾ are proportional to ã † and ã , respectively, and cos2(兩␩兩n1/2)⫽1⫺ 兩 ␩ 兩 2ã †ã , cos2(兩␩兩(n⫹1)1/2)⫽1⫺ 兩 ␩ 兩 2ãã † ). Although there is a priori no conceptual difficulty in treating this case, the corresponding dynamics becomes cumbersome when many atoms cross the cavities. We thus restrict ourself in what follows to the simpler situation in which conditions 共15兲 and 共28兲 are fulfilled.




3519

Let us determine the change of the wave function of the field mode when many atoms cross the cavities. Let ⌬t be such that the number ⌬t/ ␦ t of atoms crossing the cavities in a time interval 关 t,t⫹⌬t 兴 is large but much smaller than the inverse of the coupling strength 兩 ␩ 兩 : 1Ⰶ

⌬t Ⰶ 兩 ␩ 兩 ⫺1 , 兩 ␩ 3/2⑀ 兩 ⫺1 , 兩 ␩ ⑀ 兩 ⫺2 . ␦t

共35兲

Assuming that r g and r e are of the same order of magnitude, the numbers of atoms entering in the first cavity in state 兩 g 典 and 兩 e 典 between t and t⫹⌬t are large and approximately equal to r g ⌬t and r e ⌬t, respectively. Since the probabilities ␦ p ⫾ (t) of occurrence of jumps when one atom crosses the cavities are assumed to be as small as 兩 ␩ 兩 ⫹ 兩 ␩ 3/2⑀ 兩 ⫹ 兩 ␩ ⑀ 兩 2 , the probability of occurrence of two or more jumps between t and t⫹⌬t is negligible. Suppose first that no jump occur between these two times, i.e., that only cases 共1兲 and 共4兲 occur for all atoms. Then, 兩 ␸ 共 t⫹⌬t 兲典 ⫽¯ 共 1⫺i ␦ tK g 兲 ¯ 共 1⫺i ␦ tK e 兲 ¯ 共 1⫺i ␦ tK g 兲 ¯ 兩 ␺ 共 t 兲典 .

共36兲

There are r g ⌬t factors (1⫺i ␦ tK g ) and r e ⌬t factors (1⫺i ␦ tK e ). Let us expand the products and neglect terms of order (⌬t/ ␦ t) 2 兩 ␩ (4⫹m)/2⑀ m 兩 , m⫽0, . . . ,4, which are small by 共35兲. We find: 兩 ␺ 共 t⫹⌬t 兲典 ⫽

兩 ␸ 共 t⫹⌬t 兲典 , 储 ␸ 共 t⫹⌬t 兲储

兩 ␸ 共 t⫹⌬t 兲典 ⯝ 共 1⫺i⌬tK 兲兩 ␺ 共 t 兲典 ,

共37兲

with K⫽r g ␦ tK g ⫹r e ␦ tK e . Note that the change 兩 ␸ (t⫹⌬t) 典 ⫺ 兩 ␸ (t) 典 is proportional to ⌬t. The effective Hamiltonian K is the average of the operators K i , which describe the evolution of the quantum field as one given atom crosses the cavities without absorbing or emitting a photon, over its 共unknown兲 initial state 兩 i 典共i⫽g with probability r g ␦ t, and i⫽e with probability r e ␦ t兲. This averaging is related to our assumption that many atoms cross the cavities between t and t⫹⌬t; it is an average over the initial states of the atoms, and must be distinguished from the average over the results of the measurements. Equation 共29兲 gives K⫽

1 共 ␥ 共 ã †ã ⫹2 共 ⫺1 兲 k ⑀ *ã ⫹ 兩 ⑀ 兩 2 兲 ⫹ ␥ ⫹ 共 ãã † ⫹2 共 ⫺1 兲 k ⑀ã † ⫹ 兩 ⑀ 兩 2 兲兲 . 2i ⫺

共38兲

If a jump ⫾ occurs between t and t⫹⌬t, the change of the wave function is approximately 兩 ␺ 共 t⫹⌬t 兲典 ⫽

W ⫾兩 ␺ 共 t 兲典 . 储 W ⫾兩 ␺ 共 t 兲典储

共39兲

Actually, the change due to atoms crossing the cavities without modifying their states 关cases 共1兲 and 共4兲兴 between t and t⫹⌬t is of order (⌬t/ ␦ t)( 兩 ␩ 兩 ⫹ 兩 ␩ 3/2⑀ 兩 ). It can be neglected with respect to the change due to an atom having emitted or absorbed a photon, of order 1. Similarly, the probability ⌬p ⫾ (t) of occurrence of a jump ⫾ in the time interval 关 t,t⫹⌬t 兴 is obtained by replacing ␦ t by ⌬t in 共32兲. The above coarse-grained stochastic dynamics coincides for ␥ ⫹ ⫽r e ⫽0 with that considered by Wiseman and Milburn8 for a damped mode monitored by homodyne detection. In the Schro¨dinger picture, the mode wave function 兩 ␺ S (t⫹⌬t) 典 is given by 共39兲, in which W ⫾ is replaced by S (t)⫽e ⫺itH W ⫾ e itH , if a jump ⫾ occurs between t and t⫹⌬t. In the opposite 共and much more W⫾ probable兲 case, it is given by 共37兲, with K replaced by H⫹K S (t), K S (t)⫽e ⫺itH Ke itH . Here H ⫽ ␻ (a † a⫹1/2) is the free Hamiltonian of the field mode, and we have assumed ␻ ⌬tⰆ1. The S (t)⫽a ˜ jump operators in the Schro¨dinger picture are, up to irrelevant phase factors, W ⫺ S k ⫺i ␻ t † k i␻t ⫹(⫺1) ⑀ e and W ⫹ (t)⫽a ˜ ⫹(⫺1) ⑀ * e . Like the Hamiltonian H int , W ⫾ depends on time in the Schro¨dinger picture and is time-independent in the interaction picture. The dynamics in the


3520



Schro¨dinger picture is the same as in the interaction picture provided that ⑀ is replaced by an oscillating field ⑀ (t)⫽ ⑀ e ⫺i ␻ t . This is also true for the dynamics between jumps, K S (t) being obtained from 共38兲 by the same rule. For ⑀⫽0, the random evolution of 兩 ␺ S (t) 典 coincides with the random evolution in the model of Dalibard et al.,4 in agreement with the results of Ref. 27. For ⑀⫽0, it coincides with the QJ evolution of Sec. II with the two Lindblad operators L ⫺ ⫽ 冑␥ ⫺ã and L ⫹ ⫽ 冑␥ ⫹ã † , and with ␭ ⫺ ⫽ 冑␥ ⫺ (⫺1) k ⑀ e ⫺i ␻ t , ␭ ⫹ ⫽ 冑␥ ⫹ (⫺1) k ⑀ * e i ␻ t . All the results of this section are actually valid for an arbitrary Hamiltonian H L —describing the interaction of the two-level atoms with an arbitrary external field placed between the first cavity and the detector—such that 储 H L 储 ␶ L Ⰶ1. In fact, if we define ⑀ ⫽ 具 e 兩 e ⫺i ␶ L H L 兩 g 典 /␭ ␶ 具 e 兩 e ⫺i ␶ L H L 兩 e 典 , the only change in the calculation is the replacement of the prefactor (1⫺ 兩 ␩ ⑀ 兩 2 ) ⫺1/2 by 具 g 兩 e ⫺i ␶ L H L 兩 g 典 in 共23兲 and 共25兲 and by 具 e 兩 e ⫺i ␶ L H L 兩 e 典 in 共24兲 and 共26兲. In particular, the stochastic dynamics is not modified if the laser frequency is detuned from the atomic frequency by ␦ ⫽ ␻ L ⫺ ␻ , with 兩 ␦ 兩 ␶ L Ⰶ1. This is because ⑀ is independent of ␦ in leading order in 储 H L 储 ␶ L , i.e., the interaction time ␶ L is too short for the atoms to feel the frequency shift from the atomic transition.

C. Average dynamics of the field mode

In order to relate the random wave function dynamics with the familiar density matrix approach, let us compute the master equation satisfied by the field mode density matrix:

␳ 共 t 兲 ⫽M兩 ␺ 共 t 兲典具 ␺ 共 t 兲兩 ,

共40兲

where M denotes the mean value over the results of the measurements on the atoms 共i.e., over all quantum trajectories兲. As is well known, averaging the projector 兩 ␺ (t) 典具 ␺ (t) 兩 over all results of the measurements is the same as not performing any measurement. One may therefore equivalently determine the differential equation for the reduced density matrix ˜␳ (t) of the photon mode, in the same experimental scheme as in Fig. 1 but without the detector. The reduced density matrix ˜␳ (t) is defined as the partial trace over the atomic Hilbert spaces of the density matrix ␴ (t) of the total system ‘‘atoms⫹field mode’’ 共see Sec. II兲: ˜␳ 共 t 兲 ⫽tr共 ␴ 共 t 兲兲 .

共41兲

A

In order to justify the above-mentioned statement, let us show that the changes of ␳ (t) and ˜␳ (t) when one atom crosses the cavities are the same. The atom arrives in the first cavity in state 兩 i 典 , i⫽g or i⫽e. Before it enters in the cavity, the total density matrix ␴ (t) is a tensor product 兩 i 典具 i 兩丢 ˜␳ (t). We assume that ˜␳ (t)⫽ ␳ (t)⫽M兩 ␺ (t) 典具 ␺ (t) 兩 . It follows from the second equality in 共17兲, from 共19兲, and from the fact that the measurements on the atoms are independent:

␳ 共 t⫹ ␦ t 兲 ⫽Mt⫹ ␦ t 兩 ␺ 共 t⫹ ␦ t 兲典具 ␺ 共 t⫹ ␦ t 兲兩 ⫽tr共 Mt 兩 ⌿ ent典具 ⌿ ent兩兲 ,

共42兲

A

where 兩 ⌿ ent典 is given by 共16兲. Thus,

␳ 共 t⫹ ␦ t 兲 ⫽tr共 e ⫺i ␶ L H L e ⫺i ␶ H int兩 i 典具 i 兩 ␳ 共 t 兲 e i ␶ H inte i ␶ L H L 兲 ⫽˜␳ 共 t⫹ ␦ t 兲 ⫽tr共 e ⫺i ␶ H int␴ 共 t 兲 e i ␶ H int兲 . 共43兲 A

A

Hence the changes of ␳ (t) and ˜␳ (t) are identical as one atom crosses the cavities. The two exponentials of H L in 共43兲 disappear by cyclicity of the trace, showing that ˜␳ (t⫹ ␦ t)⫽ ␳ (t⫹ ␦ t) does not depend on the laser field E. This last point is actually clear, since if no measurement on the atoms is performed, their interaction with the laser field in the second cavity has no effect on the field in the first cavity.




3521

Since they have the same time evolution, we identify in what follows ␳ (t) and ˜␳ (t) and compute the master equation for the usual density matrix 共41兲. Two cases must be distinguished. 共1兲 The atom enters in the first cavity in its ground state: i⫽g. By replacing 共20兲 into 共43兲,

␳ 共 t⫹ ␦ t 兲 ⫽ 共 1⫺ 兩 ␩ 兩 2ã †ã 兲 1/2␳ 共 t 兲共 1⫺ 兩 ␩ 兩 2ã †ã 兲 1/2⫹ 兩 ␩ 兩 2ã ␳ 共 t 兲ã † ⬅ 共 1⫹ ␦ tLg 兲 ␳ 共 t 兲 ,

共44兲

where ã is given by 共21兲. 共2兲 The atom enters in the first cavity in its excited state: i⫽e. Then,

␳ 共 t⫹ ␦ t 兲 ⫽ 共 1⫺ 兩 ␩ 兩 2ãã † 兲 1/2␳ 共 t 兲共 1⫺ 兩 ␩ 兩 2ãã † 兲 1/2⫹ 兩 ␩ 兩 2ã † ␳ 共 t 兲ã ⬅ 共 1⫹ ␦ tLe 兲 ␳ 共 t 兲 .

共45兲

For a given initial state of the atom, the evolution in 共44兲 and 共45兲 has the general form 共7兲. Under the conditions 共15兲 and 共28兲, one has

␦ tLg 共 ␳ 兲 ⯝ 兩 ␩ 兩 2 共 ã ␳ã † ⫺ 21 兵ã †ã , ␳ 其兲 , ␦ tLe 共 ␳ 兲 ⯝ 兩 ␩ 兩共 ã ␳ã ⫺ 兵ãã , ␳ 其兲 , 2

†

1 2

†

共46兲

where the curly brackets denote the anticommutator 兵 A,B 其 ⫽AB⫹BA. Let us determine the coarse-grained evolution of ␳ (t) on the time scale ⌬t satisfying 共35兲. By the same arguments as in Sec. III B,

␳ 共 t⫹⌬t 兲 ⫽¯ 共 1⫹ ␦ tLg 兲 ¯ 共 1⫹ ␦ tLe 兲 ¯ 共 1⫹ ␦ tLg 兲 ¯ ␳ 共 t 兲 , with r g ⌬t factors (1⫹ ␦ tLg ) and r e ⌬t factors (1⫹ ␦ tLe ). One can retain only the terms of order one in ⌬t in the expansion of the product:

␳ 共 t⫹⌬t 兲 ⯝ 兵 1⫹⌬t 共 r g ␦ tLg ⫹r e ␦ tLe 兲其 ␳ 共 t 兲 .

共47兲

The superoperator inside the parentheses is the average of the superoperator Li over the initial atomic states 兩 i 典 , i⫽e or g. Writing ⌬ ␳ /⌬t⫽d␳ /dt, the coarse-grained master equation in the interaction picture is therefore d␳ ⫽ ␥ ⫺ 共 ã ␳ 共 t 兲ã † ⫺ 21 兵ã †ã , ␳ 共 t 兲其兲 ⫹ ␥ ⫹ 共 ã † ␳ 共 t 兲ã ⫺ 21 兵ãã † , ␳ 共 t 兲其兲 . dt

共48兲

Not surprisingly, this is the equation of the damped harmonic oscillator with finite temperature T. Here T is the temperature of the atomic beam:

冉

exp ⫺

冊

re ␥⫹ ␻ ⫽ ⫽ , k BT rg ␥⫺

共49兲

where k B is the Boltzmann constant. Equation 共48兲 has the general form 共1兲, with two Lindblad operators L ⫺ ⫽ 冑␥ ⫺ã and L ⫹ ⫽ 冑␥ ⫹ã † . Recall that ã coincides with the usual annihilation operator a only if k⫽0 共perturbative regime兲. In conclusion, if 兩 ␭ 兩 ␶ Ⰶ1 and condition 共28兲 holds, i.e., if each atom modifies weakly the state of the field, the average density matrix 共40兲 satisfies the master equation 共48兲. Although the quantum trajectories of the mode depend on E, the corresponding master equation is the same for all laser fields, in accordance with the general results of Sec. II. IV. SINGLE QUANTUM TRAJECTORIES OF THE FIELD MODE

We focus in this section on single quantum trajectories of the field mode, corresponding to specific realizations of the measurements. The main question addressed in the following concerns the localization properties of the random dynamics at large times, for different laser fields E. In the


3522



case E⫽0, i.e, for the ‘‘standard’’ Monte Carlo wave function dynamics, it has been shown in Ref. 27 that the field wave function evolves at large times toward Fock states 兩 n(t) 典关with a time fluctuating number of photons n(t)兴. Note that it is straightforward to prove that Fock states form an invariant family of states under the stochastic dynamics. In fact, if ⑀ ⫽0, the jump operators W ⫾ transform 兩 n 典 into 兩 n⫾1 典共up to multiplicative constants兲, and 兩 n 典 is an eigenvector of H ⫹K 0 ⫽ ␻ (n⫹1/2)⫺i( ␥ ⫺ sin2(兩␩兩n1/2)⫹ ␥ ⫹ sin2(兩␩兩(n⫹1)1/2))/(2 兩 ␩ 兩 2 ). The localization property is much more difficult to prove. For nonzero laser fields, Fock states do no longer form an invariant family and the localization is of different nature. In the case of zero temperature ( ␥ ⫹ ⫽0), it has been shown in Ref. 8 that the field mode has a diffusive dynamics in the limit ⑀ →⬁, given by a stochastic Schro¨dinger equation with real Wiener processes; we will prove in Sec. V that this remains true at any temperature. Such kind of dynamics has been widely studied in the literature.11,12,15,26,28 It is expected that localization occurs toward coherent or squeezed states if the Lindblad operators are linear combinations of a and a † . However, this has only been proved in the case L⬀(a⫹a † ) as far as we are aware.28 Here we present numerical results indicating that for T⬎0 and large enough ⑀, the mode wave function 兩 ␺ (t) 典 evolves toward squeezed states after some time ⌬␶ of the order of the thermalization time in the absence of measurements. At large times, the squeezing amplitude r(t) of the squeezed states is found to fluctuate slightly around a mean value ¯, r and the squeezing angle ␾ (t) evolves linearly in time, ␾ (t)⫽ ␾ 0 ⫺ ␻ t. Interestingly, ¯r and ␾ 0 are independent of the realization and of the initial state, being, respectively, functions of the temperature T and the laser field E only. By letting ⑀ →⬁, writing the corresponding quantum state diffusion equation 共which is derived in the general case in Sec. V兲, and using a result due to Rigo and Gisin,26 we obtain in Sec. IV B analytic expressions for ¯r and ␾ 0 in good agreement with the numerical simulations. A. Numerical results

Let us first look at the Husimi Q-function for the mode wave function in the interaction picture,

FIG. 2. Q( ␣ ,t)⫹1 for ⑀ ⫽0 and ␥ ⫹ / ␥ ⫺ ⫽3/4, at four different times: ␥ ⫺ t⫽0 共top left兲, ␥ ⫺ t⫽5 共top right兲, ␥ ⫺ t⫽20 共bottom left兲, and ␥ ⫺ t⫽60 共bottom right兲.




3523

FIG. 3. Q( ␣ ,t)⫹1 for ⑀⫽50 and ␥ ⫹ / ␥ ⫺ ⫽3/4, at the same times as in Fig. 2.

Q 共 ␣ ,t 兲 ⫽

1 兩具 ␣兩␺共 t 兲典兩2, ␲

␣ 苸C.

The simulations of single quantum trajectories for ⑀⫽0 and ⑀⫽50 give the results plotted in Figs. 2 and 3, respectively. Recall that for Fock states 兩 ␺ 典 ⫽ 兩 n 典 , Q( ␣ ) is equal to 兩 ␣ 兩 2n exp(⫺兩␣兩2)/␲n!, and for squeezed states 兩 ␺ 典 ⫽ 兩 ␣ 0 , ␰ 典 with ␰苸R, 兩 ␰ 兩 ⫽r, Q共 ␣ 兲⫽

冉

1 2ᑬ共 ␣ ⫺ ␣ 0 兲 2 2ᑣ共 ␣ ⫺ ␣ 0 兲 2 exp ⫺ ⫺ ␲ cosh共 r 兲 1⫹e ⫺2r 1⫹e 2r

冊

共see Ref. 29兲. This asymmetric paraboloid becomes symmetric for a coherent state 兩 ␺ 典 ⫽ 兩 ␣ 0 典 , which corresponds to r⫽0 in the above-mentioned formula. It is seen in Fig. 2 that in the case ⑀⫽0, the initial coherent state 兩 ␣ ⫽2⫹2i 典 is transformed into states which are close to Fock states at times t 3 ⫽20/␥ ⫺ (n⫽0) and t 4 ⫽60/␥ ⫺ (n⫽0). This is in agreement with the results of Ref. 27. On the other hand, for ⑀⫽50, Fig. 3 shows that the same initial coherent state is transformed into states which are close to squeezed states 兩 ␣ ,re i ␾ 典 . The squeezed states at t 3 and t 4 have approximately the same squeezing parameters r and ␾⫽0, but they have different centers ␣. In order to study more precisely the localization toward squeezed states at large values of ⑀, let us follow the time evolution of the mean square deviations for a given quantum trajectory t哫兩 ␺ (t) 典 : ⌬x ␾2 共 t 兲 ⫽ 具 ␺ S 共 t 兲兩 X ␾2 兩 ␺ S 共 t 兲典 ⫺ 具 ␺ S 共 t 兲兩 X ␾ 兩 ␺ S 共 t 兲典 2 , ⌬y ␾2 共 t 兲 ⫽ 具 ␺ S 共 t 兲兩 Y ␾2 兩 ␺ S 共 t 兲典 ⫺ 具 ␺ S 共 t 兲兩 Y ␾ 兩 ␺ S 共 t 兲典 2 .

共50兲

X ␾ and Y ␾ are the usual field quadrature operators rotated by an angle ␾:


3524



FIG. 4. ⌬x 2 (t) 共solid, dashed, and dotted lines兲 and ⌬x 2 (t)⌬y 2 (t) 关共⫹兲, 共〫兲, and 共䉮兲兴 vs ␥ ⫺ t for three different quantum trajectories with initial coherent state. The values of the parameters are ⑀ ⫽20 and ␥ ⫹ / ␥ ⫺ ⫽3/4. The dotted line and the triangles correspond to the exact dynamics using the nonperturbative formulas 共23兲–共26兲 for K g , W ⫾ , and K e , with 兩 ␭ 兩 ␶ ⫽0.93⫻10⫺2 .

X ␾⫽

ae ⫺i ␾ ⫹a † e i ␾ , 2

Y ␾ ⫽X ␾ ⫹ ␲ /2⫽

ae ⫺i ␾ ⫺a † e i ␾ . 2i

共51兲

We denote by ␾ min(t) the angle for which ⌬x ␾2 (t) is minimum. The minimum mean square deviation ⌬x ␾2 (t)(t) is denoted by ⌬x 2 (t) and, similarly, ⌬y ␾2 (t)(t) is denoted by ⌬y 2 (t). Obmin min viously, ⌬x 2 (t) and ⌬y 2 (t) remain unchanged if the wave function 兩 ␺ S (t) 典 in the Schro¨dinger picture is replaced in 共50兲 by the wave function 兩 ␺ (t) 典 in the interaction picture. This replacement leads to an increase of ␾ min(t) by ␻ t. The time evolutions of ⌬x 2 (t) and ⌬x 2 (t)⌬y 2 (t) are shown in Figs. 4 and 5, for different quantum trajectories starting from the same coherent state 兩 ␺ (0) 典 ⫽ 兩 ␣ 典 . Note that the time scale is of order of the thermalization time of the density matrix ␳ (t). Figure 4 corresponds to ⑀ ⫽20 and Fig. 5 to ⑀ ⫽100. One sees in both Figs. 4 and 5 that ⌬x 2 (t) begins to fluctuate around a mean

FIG. 5. Same as in Fig. 4 but with ⑀ ⫽100 共the nonperturbative results are not shown but look similar兲. In the upper square, ⌬x 2 (t) and ⌬x 2 (t)⌬y 2 (t) are shown on a finer time scale, on which discontinuous quantum jumps can be seen separately.




3525

FIG. 6. Time averages ⌬x 2 共plain line兲 and ⌬x 2 ⌬y 2 共〫兲 for different values of ᑬ⑀ 共shown in the horizontal axis兲 and ᑣ⑀⫽0. Inset: same for different values of arg ⑀ 共shown in degrees in the horizontal axis兲 and fixed modulus 兩⑀兩⫽50. The time average is taken on the interval 关 10/␥ ⫺ ,100/␥ ⫺ 兴 and ␥ ⫹ / ␥ ⫺ ⫽3/4.

value ⌬x 2 after some transient time ⌬ ␶ ⯝10/␥ ⫺ . The comparison of the numerical results for the different trajectories shows that ⌬x 2 does not depend upon the specific realization. The fluctuations are considerably reduced in the case ⑀⫽100 共Fig. 5兲 with respect to the case ⑀⫽20 共Fig. 4兲. Moreover, in the former case, the product ⌬x 2 (t)⌬y 2 (t) is much closer to the minimum value 1/16 allowed by the Heisenberg uncertainty principle. The variation of ⌬x 2 and ⌬x 2 ⌬y 2 with ⑀ for fixed temperature T⯝0.288␻ /k B is presented in Fig. 6. One observes that ⌬x 2 is almost constant for 50⭐⑀⭐100. Furthermore, the time average of ⌬x 2 (t)⌬y 2 (t) goes closer and closer to 1/16 as ⑀ increases. The result presented in the inset shows moreover that ⌬x 2 and ⌬x 2 ⌬y 2 are independent of arg(⑀), up to small fluctuations. Most numerical simulations were done using the approximations 共31兲 and 共38兲 for the jump operators W ⫾ and for K, together with ã ⫽a. This is justified if the maximum number of photons 3/2 Ⰶ 兩 ␩ 兩 ⫺4 兩 ⑀ 兩 ⫺1 , n max is small with respect to 兩 ␩ 兩 ⫺2 ⫽ 兩 ␭ 兩 ⫺2 ␶ ⫺2 , and 兩 ␩ 兩 ⫺2 ⫽ 兩 ␭ 兩 ⫺2 ␶ ⫺2 , and n max 3/2 ⫺2 n maxⰆ 兩 ␩ 兩兩 ⑀ 兩共see Sec. III B兲. Since we worked in the Fock states basis, n max was bounded by the dimension of the Hilbert space 共which was taken between 75 and 150兲. One may be worried, however, that, despite the smallness of the error made at each time step ␦ t, the error made after many steps might be large and could invalidate our results in the long time limit. We checked that this is actually not the case by integrating numerically the exact dynamics, using formulas 共23兲– 共26兲 for the evolution on each time step ␦ t, and the exact possibilities of occurrence of the four cases. The curve in the dotted line and the triangles in Fig. 4 are the values of ⌬x 2 (t) and ⌬x 2 (t)⌬y 2 (t) obtained from a simulation of this nonperturbative dynamics. This means using the exact formulas 共23兲–共26兲 for the evolution on each time step ␦ t, and the exact probabilities of occurrence of the four cases. The total number of atoms crossing the cavity in the whole time interval is 2⫻106 , and 兩 ␩ 兩 ⫽ 冑( ␥ ⫺ ⫹ ␥ ⫹ ) ␦ t⯝0.0093. The exact results show very similar fluctuations, around the same values ⌬x 2 and ⌬x 2 ⌬y 2 , as the two trajectories obtained using the perturbative scheme. No systematic deviation increasing with time is seen. This result, together with other simulations for different values of ⑀ and ␩,31 shows errors in the considered time range. Note however that if 兩 ␩ 兩 is too large 共for values in the range 0.06 –0.09 or larger in Fig. 4, and 0.02–0.04 or larger in Fig. 5兲, large time fluctuations of ⌬x 2 (t) and ⌬x 2 (t)⌬y 2 (t) are observed and therefore the localization is of different nature. In all our simulations, we found that ⌬x 2 and ⌬x 2 ⌬y 2 are insensitive to the initial state 兩␺共0兲典 of the field mode. This is illustrated in Fig. 7 for ⑀⫽20 and ⑀ ⫽20⫹10i 共similar results are


3526



FIG. 7. ⌬x 2 (t) 共lines兲 and ⌬x 2 (t)⌬y 2 (t) 关共⫹兲, 共〫兲, and 共䊊兲兴 vs ␥ ⫺ t for trajectories with different initial states 兩␺共0兲典. Left-hand side: in the two shown trajectories, 具 n 兩 ␺ (0) 典 is chosen randomly for 0⭐n⭐20 and vanishes for n⬎20; 兩␺共0兲典 and the realization of the measurements are different in each trajectory; ⑀ ⫽20⫹10i and ␥ ⫹ / ␥ ⫺ ⫽1/2. Right-hand side: 共1兲兩␺共0典⫽兩␣典 with ␣ ⫽ 冑3/2(1⫹i) 关plain line, (䊊);兴共2兲 id. with ␣ ⫽⫺1⫹5i 关dashed line, 共⫹兲;兴共3兲兩 ␺ (0) 典 ⫽ 兩 n⫽5 典关dotted line, 共〫兲兴; in all cases, ⑀⫽20 and ␥ ⫹ / ␥ ⫺ ⫽3/4. The dotted line on the left-hand side and dot-dashed line on the right-hand side correspond to the theoretical result 共64兲 for ⑀ →⬁.

obtained for ⑀⫽100兲. On the right-hand side of the Fig. 7, it is seen that ⌬x 2 (t) differs noticeably at small t’s if 兩␺共0兲典 is a coherent state 关cases 共1兲 and 共2兲兴 and if it is a Fock state 关case 共3兲; the first small time values are outside the range of the figure: ⌬x 2 (0)⫽2.75兴. However, the three curves are hard to distinguish for ␥ ⫺ tⲏ10. In the two trajectories shown on the left-hand side, the two distinct initial states are chosen by picking randomly the 20 first components 具 n 兩 ␺ (0) 典 in the Fock states basis 共the realization of the measurements is also different in the two cases兲. For times bigger than 10/␥ ⫺ , one observes that ⌬x 2 (t) fluctuates around the same value for both trajectories. For the quantum trajectories studied in Figs. 3–5, the angle ␾ min(t)⫹␻t is found to be zero at times t⭓⌬ ␶ . Nonvanishing angles are obtained if one considers non real ⑀’s. The time dependence of ␾ min(t)⫹␻t for ⑀ ⫽20⫹10i is presented in Fig. 8 for different initial states. It is seen that it evolves toward a constant value ␾ 0 which neither depends on 兩␺共0兲典 nor on the ratio ␥ ⫹ / ␥ ⫺ . As shown in the inset, we obtain ␾ 0 ⫽arg(⑀).

FIG. 8. Angle ␾ min(t)⫹␻t 共in degrees兲 vs ␥ ⫺ t for ⑀ ⫽20⫹10i and different initial states and temperatures: 共1兲兩␺共0兲典⫽兩␣典 with ␣ ⫽2⫺2i; ␥ ⫹ / ␥ ⫺ ⫽1/4 共䉭兲; 共2兲兩 ␺ (0) 典 ⫽(2⫹2e ⫺9 ) ⫺1/2 共兩␣典⫹兩␤典兲 with ␣ ⫽⫺3i and ␤⫽3; ␥ ⫹ / ␥ ⫺ ⫽3/4 共䊉兲; 共3兲兩 ␺ (0) 典 ⫽ 兩 n 典 with n⫽10; ␥ ⫹ / ␥ ⫺ ⫽3/4 共⫹兲; 共4兲兩␺共0兲典 is chosen randomly as in Fig. 7; ␥ ⫹ / ␥ ⫺ ⫽1/2 共dashed line兲. Inset: time average of ␾ 0 (t)⫽ ␾ min(t)⫹␻t as function of arg ⑀ 共both in degrees兲 for 兩⑀兩⫽50 and ␥ ⫹ / ␥ ⫺ ⫽3/4 共time average as in ¯ ⫽arg(⑀) mod. 180. Fig. 6兲. The result is well fitted by the broken lines ␾ 0




3527

FIG. 9. Time average of the probability 兩具 n 兩 ␺ (t) 典兩 2 to find n photons for ⑀⫽50, ␥ ⫹ / ␥ ⫺ ⫽3/4 共⫹兲. The average is taken over 5000 discrete times in the interval 关 0,100/␥ ⫺ 兴 . The solid line corresponds to the Bose–Einstein distribution.

Bringing together the above-mentioned results, we are led to the following conclusions. For large 兩⑀兩 and small 兩␩兩 smaller than ⯝0.01, the wave function 兩 ␺ S (t) 典 of the field mode evolves, whatever its initial state 兩 ␺ S (0) 典 , to some almost minimum uncertainty states 共MUS兲 at times t ⭓⌬ ␶ , where ⌬␶ is a transient time. Moreover, if T⬎0, ⌬x 2 共 t 兲 ⯝⌬x 2 ⬍1/4,

␾ min共 t 兲 ⯝ ␾ 0 ⫺ ␻ t

共52兲

at large times where both ⌬x 2 and ␾ 0 ⫽arg(⑀) are independent of 兩 ␺ S (0) 典 and of the realization of the measurements. Therefore, provided 兩⑀兩 is big enough and 兩␩兩 small enough, the state of the field mode at times t⭓⌬ ␶ is close to a squeezed state: 兩 ␺ S共 t 兲典 ⯝ 兩 ␣ 共 t 兲 , ␰ 共 t 兲典 .

共53兲

The squeezing amplitude r(t)⫽ 兩 ␰ (t) 兩 fluctuates slightly around a time-independent value r⯝⫺ln(4⌬x2)/2 and arg(␰(t))⯝2(␾0⫺␻t). The center ␣ (t) of the squeezed state moves around in the complex plane. Actually, by ergodicity, the time average of 兩 ␺ S (t) 典具 ␺ S (t) 兩 must coincide with the equilibrium density matrix ␳ (eq) ⫽Z ⫺1 exp(⫺␻(a†a⫹1/2)/k B T). To check ergodicity, we have computed numerically the time average of 兩具 n 兩 ␺ (t) 典兩 2 on the interval 关 0,100/␥ ⫺ 兴 for a single quantum trajectory. It is indeed seen in Fig. 9 that it reproduces well the Bose–Einstein exponen共eq兲 . tial distribution ␳ nn B. Analytical results

The above-given numerical results suggest that the dynamics has the localization property toward squeezed states in the limit of large laser fields 兩⑀兩→⬁. In particular, the squeezed states should form an invariant family of states under the stochastic dynamics in this limit. The second statement can be shown analytically as follows. We restrict our analysis here to the perturbative regime where 兩␩兩Ⰶ1, 兩␩⑀兩Ⰶ1 and 共28兲 holds with k⫽0, so that ã ⯝a. As said previously, one can describe the mode’s dynamics in the limit 兩⑀兩→⬁, 兩␩兩→0, 兩␩⑀兩→0 by a stochastic Schro¨dinger equation with real Wiener processes 共quantum state diffusion兲. This is because the probability of occurrence of jumps grows like 兩 ⑀ 兩 2 共for instance, in Figs. 4 and 5, the total number of jumps in the whole time interval 关 0,100/␥ ⫺ 兴 is close to 7⫻104 and 1.75⫻106 , respectively兲. On the other hand, as is clear from 共31兲 and 共39兲, the change of the wave function during a jump is of order 1/⑀. Hence there are infinitely many jumps with an infinitesimal impact on the wave function in the


3528



limit 兩⑀兩→⬁. On a time scale much bigger than the inverse frequencies of jumps 关but small enough so that 兩 ␺ (t) 典 does not change much on such a time scale兴, 兩 ␺ (t) 典 satisfies the following Itoˆ stochastic differential equation in the limit ⑀→⬁, arg ⑀⫽␪: 兩 d␺ 典 ⫽

冋冑

␥ ⫺ 共 e ⫺i ␪ a⫺ᑬ具 e ⫺i ␪ a 典 t 兲 dw ⫺ 共 t 兲 ⫹ 冑␥ ⫹ 共 e i ␪ a † ⫺ᑬ具 e i ␪ a † 典 t 兲 dw ⫹ 共 t 兲

冉

⫹ᑬ具 e i ␪ a † 典 t ␥ ⫺ e ⫺i ␪ a⫹ ␥ ⫹ e i ␪ a † ⫺

冊

册

␥ ⫺⫹ ␥ ⫹ ᑬ具 e i ␪ a † 典 t dt⫺iK 0 dt 兩 ␺ 共 t 兲典 . 2

共54兲

The notation is as follows: K 0 ⫽( ␥ ⫺ a † a⫹ ␥ ⫹ aa † )/2i, 具 O 典 t ⫽ 具 ␺ (t) 兩 O 兩 ␺ (t) 典 is the quantum expectation at time t and dw ⫾ (t) are the stochastic infinitesimal increments of two independent real Wiener processes, which have zero mean and satisfy the Itoˆ rules:30 dw ⫾ 共 t 兲 dw ⫾ 共 t 兲 ⫽dt,

dw ⫺ 共 t 兲 dw ⫹ 共 t 兲 ⫽0,

dw ⫾ 共 t 兲 dt⫽0.

共55兲

Equation 共54兲 will be derived in Sec. V in the general case. It belongs to the class of stochastic Schro¨dinger equations studied in Refs. 11 and 12 and has been derived from the QJ dynamics by Wiseman and Milburn8 共see also Refs. 5 and 9兲 in the case of a mode in a decaying cavity 共corresponding here to ␥ ⫹ ⫽0兲. It can actually be derived directly from the stochastic dynamics of Sec. III B, Eqs. 共23兲–共26兲, with the probabilities 共27兲, in the limit 兩⑀兩→⬁, 兩␩兩→0, 兩⑀␩兩⫽const., under assumption 共28兲 with k⫽0.31 A related equation with complex Wiener processes has been studied in Ref. 10. Rigo and Gisin26 have shown that the stochastic Schro¨dinger equation 共54兲 preserves squeezed states. Since 共54兲 actually differs from the equation considered by these authors by some additional phase factors e ⫾i ␪ , and the explicit solution of the evolution equations for the squeezing parameters is not given in Ref. 26, we briefly recall here their derivation. We use the Itoˆ formalism of stochastic differential equations,30 whereas Stratanovich formalism was used in Ref. 26. It is convenient to characterize squeezed states by the following criterion: 兩 ␺ 典 ⫽ 储 ␺ 储兩 ␣ , ␰ ⫽re 2i ␾ 典 ⇔ 共 a⫺⌫a † ⫺ ␤ 兲兩 ␺ 典 ⫽0,

⌫⫽⫺e 2i ␾ tanh共 r 兲 ,

␤ ⫽ ␣ ⫺⌫ ␣ * .

共56兲

The family of the squeezed states is invariant under 共54兲 if 兩 ␺ 典 ⫹ 兩 d ␺ 典 remains a squeezed state for any 兩␺典⫽兩␣,␰典, i.e., (a⫺⌫a † ⫺d⌫ a † ⫺ ␤ ⫺d␤ )( 兩 ␺ 典 ⫹ 兩 d␺ 典 )⫽0. This is equivalent to26 关 a⫺⌫a † ,D 共 ␺ 兲兴兩 ␺ 典 ⫺ 共 d⌫ a † ⫹d␤ 兲 D 共 ␺ 兲兩 ␺ 典 ⫽ 共 d⌫ a † ⫹d␤ 兲兩 ␺ 典 ,

共57兲

where D( ␺ ) is the operator inside the square brackets in 共54兲. The left-hand side is found to be

冉

⫺

␥ ⫹⫹ ␥ ⫺ 共 a⫹⌫ a † 兲 dt⫹ᑬ具 e i ␪ a † 典共 ␥ ⫹ e i ␪ dt⫹ ␥ ⫺ e ⫺i ␪ ⌫ dt⫹ 冑␥ ⫹ d␤ dw ⫹ ⫹ 冑␥ ⫺ d␤ dw ⫺ 兲 2

冊

⫹ 冑␥ ⫹ e i ␪ 共 ⫺a † d␤ dw ⫹ ⫹dw ⫹ 兲 ⫹ 冑␥ ⫺ e ⫺i ␪ 共 ⫺a d␤ dw ⫺ ⫹⌫ dw ⫺ 兲兩 ␺ 典 .

共58兲

We have thrown away all terms d⌫ dw ⫾ since d⌫ is proportional to dt. This is because, by inspection of 共58兲, the terms containing the noises dw ⫾ on the left-hand side of 共57兲 are proportional to 兩 ␺ 典 . Multiplying both members of 共57兲 by dw ⫾ and using 共55兲 and 共58兲 gives d␤ dw ⫹ ⫽ 冑␥ ⫹ e i ␪ dt,

d␤ dw ⫺ ⫽ 冑␥ ⫺ e ⫺i ␪ ⌫ dt.

共59兲

Now we use the well-known identity S(⫺ ␰ )D(⫺ ␣ )aD( ␣ )S( ␰ )⫽a cosh r⫹⌫ a† cosh r⫹␣, where S( ␰ )⫽exp(␰*a2/2⫺ ␰ a †2 /2) and D( ␣ )⫽exp(␣a†⫺␣*a) are, respectively, the squeezing and the displacement operators.2 The squeezed state 兩 ␺ 典 is equal to D( ␣ )S( ␰ ) 兩 0 典 . Let us multiply the two members of 共57兲 by S(⫺ ␰ )D(⫺ ␣ ) and substitute 共58兲 and 共59兲 into this equation. This leads to the two coupled stochastic differential equations:




d⌫⫽⫺e 2i ␪ 共 1⫹e ⫺2i ␪ ⌫ 兲共 ␥ ⫹ ⫹ ␥ ⫺ e ⫺2i ␪ ⌫ 兲 dt,

冉

冊

␥ ⫺⫹ ␥ ⫹ ⫹ ␥ ⫺ e ⫺2i ␪ ⌫ ␤ dt⫹2ᑬ共 e i ␪ a * 兲共 ␥ ⫹ e i ␪ ⫹ ␥ ⫺ e ⫺i ␪ ⌫ 兲 dt d␤ ⫽⫺ 2

3529

共60兲

⫹ 冑␥ ⫹ e i ␪ dw ⫹ ⫹ 冑␥ ⫺ e ⫺i ␪ ⌫ dw ⫺ . These equations are equivalent to 共57兲 and have a solution. Hence the squeezed states 兩␣,␰典 form an invariant family under the quantum state diffusion dynamics, and ␣ and ␰ evolve in time according to 共60兲. We now obtain the squeezing amplitudes r(t) and angles ␾ (t) of the squeezed states by solving 共60兲. They are determined by the first equation: ⌫ 共 t 兲 ⫽⫺e 2i ␪

␥ ⫹ e ( ␥ ⫺ ⫺ ␥ ⫹ )t ⫹c ⫽⫺e 2i ␾ (t) tanh共 r 共 t 兲兲 , ␥ ⫺ e ( ␥ ⫺ ⫺ ␥ ⫹ )t ⫹c

共61兲

where c is an arbitrary complex constant. Thus r(t) and ␾ (t) are deterministic, unlike ␣ (t), which is given by the second equation. Going back to the Schro¨dinger picture, one has for t( ␥ ⫺ ⫺ ␥ ⫹ )Ⰷ1: tanh共 r 共 t 兲兲 ⯝

冉

冊

␥⫹ ␻ ⫽exp ⫺ , ␥⫺ k BT

␾ 共 t 兲 ⯝ ␪ ⫺ ␻ t.

共62兲

It is worth noticing that perfect squeezing 关 r(t)→⬁ 兴 is obtained in the high temperature limit k B TⰇ ␻ . The rate ( ␥ ⫺ ⫺ ␥ ⫹ ) ⫺1 of convergence of tanh r(t) to tanh ¯⫽exp(⫺ r ␻/kBT) also tends to infinity in this limit. Since the number of photons in the cavity becomes very large at very high temperatures and long times, the perturbative approximation 共28兲 should however break down at some point. This can put a limitation on the attainment of very large r(t). A more detailed study of this apparently surprising result is the object in a separate work.31 As stated previously, the second equation in 共62兲 agrees quite well with the results of the numerical simulations of Fig. 8. Using the minimum mean square deviation ⌬x 2 ⫽e ⫺2r /4 of squeezed states2 yields the time average ⌬x 2 : ⌬x 2 ⯝

␥ ⫺⫺ ␥ ⫹ . 4共 ␥ ⫺⫹ ␥ ⫹ 兲

共63兲

In the particular case of an initial coherent state 兩 ␺ (0) 典 ⫽ 兩 ␣ 典 , the constant c is equal to ⫺ ␥ ⫹ , so that 共62兲 is exact at all times t⭓0 and ⌬x 2 共 t 兲 ⯝

␥ ⫺⫺ ␥ ⫹ . 4 ␥ ⫺ ⫹4 ␥ ⫹ 共 1⫺2e ⫺( ␥ ⫺ ⫺ ␥ ⫹ )t 兲

共64兲

This solution is compared in Fig. 7 with the numerical results for 兩 ⑀ 兩 ⬃20. The exact value of ⌬x 2 is close to the approximated value 1/12 obtained from 共63兲 for ␥ ⫹ / ␥ ⫺ ⫽1/2 共left-hand side of Fig. 7兲. It is a bit higher than the theoretical prediction 1/28⯝0.0357 for ␥ ⫹ / ␥ ⫺ ⫽3/4 共right-hand side兲. For 50⭐⑀⭐100, one has a better agreement, as seen in Fig. 6. Hence the dynamics of the mode is well described by the quantum state diffusion equation 共54兲 for ⑀ ⭓50, at this temperature. ¨ DINGER EQUATIONS V. STOCHASTIC SCHRO

In this section we discuss the mathematical link between the quantum jump dynamics of Sec. II and various stochastic Schro¨dinger equations found in the literature. The analysis is done for arbitrary open quantum systems having a Lindblad-type dynamics.


3530



A. Linear quantum jump dynamics

A linear version of the QJ dynamics of Sec. II, in which the random wave function is not normalized at each time step ␦ t, has been introduced in Ref. 18. The unormalized wave function 兩 ␸ (t) 典 of the open system S satisfies the following Itoˆ stochastic differential equation:

冋

兩 d␸ 典 ⫽ ⫺i 共 H⫹K 兲 dt⫹

册

兺m 共 W m ⫺1 兲 dN m共 t 兲兩 ␸ 共 t 兲典 ,

共65兲

where H is the Hamiltonian of S, the jump operators W m are related to the Lindblad operators L m by L m ⫽ 冑␥ m 共 W m ⫺1 兲 ,

共66兲

and dN m (t)⫽0,1 are the stochastic increments of independent Poisson processes N m with parameters ␥ m . These increments have mean M dN m (t)⫽ ␥ m dt and satisfy the Itoˆ rules: dN m 共 t 兲 dN n 共 t 兲 ⫽ ␦ n,m dN m 共 t 兲 ,

共67兲

dN m 共 t 兲 dt⫽0.

Equation 共65兲 has been first considered by Belavkin14 with some jump operators W m proportional to L m 关the relation with the jump operators 共66兲 is given by the transformation 共8兲兴. It has also been considered independently in Ref. 16. It is easy to show that ␳ (t)⫽M兩 ␸ (t) 典具 ␸ (t) 兩 obeys the Lindblad equation 共1兲 if K is appropriately chosen. In fact, by 共65兲 and 共67兲, d␳ ⫽M共兩 d␸ 典具 ␸ 兩 ⫹ 兩 ␸ 典具 d␸ 兩 ⫹ 兩 d␸ 典具 d␸ 兩兲 ⫽⫺i 关 H, ␳ 兴 dt⫺iK ␳ dt⫹i ␳ K † dt⫹

兺m ␥ m共共 W m ⫺1 兲 ␳ ⫹ ␳ 共 W m† ⫺1 兲 ⫹ 共 W m ⫺1 兲 ␳ 共 W m† ⫺1 兲兲 dt.

Replacing 共66兲 into this equation yields d␳ ⫽⫺i 关 H, ␳ 兴 dt⫹

兺m

冉

冊

1 † † L m␳ L m ⫺ 兵Lm L m , ␳ 其 dt, 2

provided that K⫽

1 2i

1

兺m ␥ m共 W m† ⫹1 兲共 W m ⫺1 兲 ⫽ 2i 兺m 共 L m† L m ⫹2 冑␥ m L m 兲 .

共68兲

To give a physical meaning to the linear stochastic dynamics, one must choose the jump rates ␥ m equal to the probability per unit time of the corresponding transitions. These are given by the Fermi golden rule to second order in perturbation theory. For instance, for the field mode considered in Sec. III, ␥ ⫾ are the damping rates 共33兲 for the absorption and emission of a photon by the atoms. Let us write 共65兲 in the ‘‘dissipative interaction picture’’ 兩 ˜␸ (t) 典 ⫽U(t) 兩 ␸ (t) 典 , where U(t) ˜ m (t)⫺1)dN m (t) 兩 ˜␸ (t) 典 , with W ˜ m (t)⫽U(t)W m U(⫺t). ⫽exp(it(H⫹K)). It reads 兩 d˜␸ 典 ⫽ 兺 m (W This implies: ˜ m 共 t p 兲 ¯W ˜ m 共 t 1 兲兩 ␸共 0 兲典 , 兩˜ ␸ 共 t 兲典 ⫽W p 1 where 0⭐t 1 ⭐¯⭐t p ⭐¯ are the jump times 共times such that 兺 m dN m (t)⫽1兲, t p ⭐t⬍t p⫹1 , and m q is the index of the jump occurring at time t q 共dN m q (t q )⫽1, q⫽1,...,p兲. Hence the stochastic Schro¨dinger equation 共65兲 admits the solution: 兩 ␸ 共 t 兲典 ⫽e ⫺i(t⫺t p )(H⫹K) W m p e ⫺i(t p ⫺t p⫺1 )(H⫹K) ¯W m 1 e ⫺it 1 (H⫹K) 兩 ␸ 共 0 兲典 ,

t p ⭐t⬍t p⫹1 . 共69兲




3531

In other words, the evolution of the quantum state may be computed as follows. 共1兲 If there is a jump m between t and t⫹dt, then 兩 ␸ 共 t⫹dt 兲典 ⫽W m 兩 ␸ 共 t 兲典 .

共70兲

This occurs with a probability ␥ m dt independent of 兩 ␸ (t) 典 . 共2兲 If no jump occurs between t and t⫹dt, then 兩 ␸ 共 t⫹dt 兲典 ⫽ 共 1⫺i dt 共 H⫹K 兲兲兩 ␸ 共 t 兲典 .

共71兲

If ␥ ⫽ 兺 m ␥ m ⬍⬁, the time delays t⫺t p ,t p ⫺t p⫺1 ,...,t 1 and the jump indices m p ,...,m 1 in 共69兲 are independent random variables. The time delays are equally distributed, according to the exponential law e ⫺ ␥ s ds. Because of the independence of the Poisson processes N m , the probability that m q ⫽m is ␥ m / ␥ . It has been proven in Ref. 18 that, under appropriate hypothesis on W m and ␥ m , the wave function 共69兲 is still well defined 共as some limit of the right-hand side兲 if ␥ ⫽⬁, in which case infinitely many jumps occur between 0 and t. This result is important for electrons in strongly disordered solids, where m⫽(i, j) labels pairs of eigenstates 兩 i 典 , 兩 j 典 of the electronic Hamiltonian, and L i j is equal or ‘‘close’’ to 冑␥ i j 兩 j 典具 i 兩 , i.e., W i j ⯝1⫹ 兩 j 典具 i 兩共locality condition兲. Then, the number of jumps in a finite interval becomes infinite in the infinite volume limit, due to the divergence of the double sum 兺 i, j ␥ i j . In this case, the presence of the identity operator inside the parentheses in 共66兲 is necessary for the mathematical definiteness of 兩 ␸ (t) 典 . B. Nonlinear quantum jump dynamics

The nonlinear QJ scheme of Sec. II can be deduced from the above-mentioned linear QJ dynamics in the following way. By comparing 共9兲 and 共4兲 with 共70兲 and 共71兲, it is clear that, for a given realization of the jumps, the normalized wave function 兩␺共 t 兲典⫽

兩␸共 t 兲典储 ␸ 共 t 兲储

共72兲

evolves according to the nonlinear QJ scheme with ␭ m ⫽ 冑␥ m . However, it was argued in Sec. II that the density matrix ␳ (t) is the mean value of 兩 ␺ (t) 典具 ␺ (t) 兩 , whereas it has been seen earlier that ␳ (t)⫽M兩 ␸ (t) 典具 ␸ (t) 兩 . This means that the probability P ⬘ attached to 兩 ␺ (t) 典 is different from the probability P attached to 兩 ␸ (t) 典 , that is, to the Poisson processes N m . This change of probability P→ P ⬘ provides the link between the two random evolutions for 兩 ␸ (t) 典 and 兩 ␺ (t) 典 . We define it as follows. Let us denote by M⬘ and M the mean values with respect to P ⬘ and P, respectively. Let F be an arbitrary 共operator-valued兲 stochastic process such that F(t) depends only upon the realizations of the jump times up to time t. Such F is said to be adapted to the filtration of the Poisson processes N m . 33 We ask that M⬘ 共 F 共 t 兲兲 ⫽M共 F 共 t 兲储 ␸ 共 t 兲储 2 兲

共73兲

for any such process F. Taking F(t)⫽ 兩 ␺ (t) 典具 ␺ (t) 兩 , this implies in particular

␳ 共 t 兲 ⫽M⬘ 兩 ␺ 共 t 兲典具 ␺ 共 t 兲兩 ⫽M兩 ␸ 共 t 兲典具 ␸ 共 t 兲兩 .

共74兲

Let us compute the probability of occurrence of a jump m between times t and t⫹ ␦ t for the new probability P ⬘ :

␦ p m 共 t 兲 ⫽M⬘ 共 ␦ N m 共 t 兲兩 ␺ 共 t 兲兲 .

共75兲

The right-hand side is the conditional 共mean兲 expectation of ␦ N m (t) given 兩 ␺ (t) 典 , for the probability P ⬘ . Indeed, the ( P ⬘ ) probability of a jump between t and t⫹ ␦ t depends upon the wave function 兩 ␺ (t) 典 at time t. Let F(t) be an arbitrary stochastic force adapted to the filtration of the


3532



N m ’s. Then F(t) ␦ N m (t) depends upon the realizations of the N m ’s until time t⫹ ␦ t 关recall that ␦ N m (t)⫽N m (t⫹ ␦ t)⫺N m (t)]. Therefore, replacing F(t) by F(t) ␦ N m (t) in 共73兲, M⬘ 共 F 共 t 兲 ␦ N m 共 t 兲兲 ⫽M共 F 共 t 兲 ␦ N m 共 t 兲储 ␸ 共 t⫹ ␦ t 兲储 2 兲 ⫽M共 F 共 t 兲 ␦ N m 共 t 兲储 W m 兩 ␸ 共 t 兲典储 2 ). Formula 共70兲 together with the fact that ␦ N m (t)⫽0,1 have been used in the second line. By the independence of the forward increment ␦ N m (t) and F(t) 储 W m 兩 ␸ (t) 典储 2 , one gets M⬘ 共 F 共 t 兲 ␦ N m 共 t 兲兲 ⫽M共 ␦ N m 共 t 兲兲 M共 F 共 t 兲储 W m 兩 ␸ 共 t 兲典储 2 )⫽ ␥ m ␦ tM⬘ 共 F 共 t 兲储 W m 兩 ␸ 共 t 兲典储 2 储 ␸ 共 t 兲储 ⫺2 ). 共76兲 But F(t) is arbitrary, thus 共76兲 implies the identity of the conditional expectations:

␦ p m 共 t 兲 ⫽M⬘ 共 ␦ N m 共 t 兲兩 ␺ 共 t 兲兲 ⫽ ␥ m ␦ tM⬘ 共储 W m 兩 ␸ 共 t 兲典储 2 储兩 ␸ 共 t 兲典储 ⫺2 兩 ␺ 共 t 兲 )⫽ ␥ m ␦ t 储 W m 兩 ␺ 共 t 兲典储 2 ,

共77兲

in accordance with 共12兲. As a result, the stochastic evolution of the normalized wave function 兩 ␺ (t) 典 with probability P ⬘ coincides with that described in Sec. II for ␭ m ⫽ 冑␥ m . It is moreover given by the nonlinear stochastic Schro¨dinger equation:15

冋冉

冊

1 兩 d␺ 典 ⫽ ⫺i H⫹K⫹ 具 K † ⫺K 典 t dt⫹ 2

冉

W

兺m 冑具 W † mW 典 ⫺1 m m t

冊册

dN m 共 t 兲兩 ␺ 共 t 兲典 ,

共78兲

with 具 O 典 t ⫽ 具 ␺ (t) 兩 O 兩 ␺ (t) 典 . This equation is readily obtained by computing 兩 ␺ (t⫹dt) 典 from 共70兲 and 共71兲. C. Linear quantum state diffusion

It has been shown by Carmichael and Wiseman and Milburn5,8 that the nonlinear QJ dynamics of the quantum field considered in Sec. III, with ␥ ⫹ ⫽0, is well described in limit 兩⑀兩→⬁ by a quantum state diffusion 共QSD兲 stochastic equation involving real Wiener processes 共white noise兲. The linear version of QSD is obtained in this section in the more general setting of arbitrary Markovian quantum open systems, by taking the limit of infinite jump rates ␥ m →⬁ in the linear QJ dynamics of Sec. V A. Following Wiseman and Milburn,8 we introduce a small dimensionless parameter ␧⬎0 that will tend to zero. Our goal is to increase up to infinity the rates ␥ m of the jumps in the linear QJ dynamics, without modifying the master equation giving the average dynamics. Hence the Lindblad operators L m are here considered as fixed, i.e., independent of ␧. So are the damping rates contained in the master equation, given by some ␧-independent rates ¯␥ m ⬎0. The jump rates ␥ m are assumed to go to infinity like ␧ ⫺4 :

␥ m ⫽␧ ⫺4¯␥ m .

共79兲

The magnitude of the negative power of ␧ is chosen for future convenience. Let 兩 ␸ (t) 典 be the solution of the linear QJ stochastic equation 共65兲. We are interested in the variation of 兩 ␸ (t) 典 on ⫺1 such that infinitely many jumps occur between t and t⫹⌬t in the limit a time interval ⌬tⰇ ␥ m ␧→0. On the other hand, we want ⌬t to be small enough so that the change 兩 ⌬ ␸ 典 ⫽ 兩 ␸ (t⫹⌬t) 典 ⫺ 兩 ␸ (t) 典 of the wave function goes to zero as ␧→0. A possible ⌬t realizing these two conditions is ⌬t⫽␧ 3¯␥ ⫺1 ,

共80兲

where ¯␥ ⫽ 兺 m ¯␥ m is the sum of the fixed damping rates. Indeed, from 共66兲, ⫺1/2 Lm . W m ⫽1⫹␧ 2¯␥ m

共81兲




3533

Since the L m are ␧-independent, the impact of each jump on 兩␸典 is of order ␧ 2 . Moreover, the number of jumps m between t and t⫹⌬t is of order ␥ m ⌬t⫽␧ ⫺1¯␥ m / ¯␥ . This means that the impact of the jumps between t and t⫹⌬t is of order ␧ ⫺1 ⫻␧ 2 ⫽␧, which is indeed small for small ␧. Let ⌬N m (t) be the number of jumps m in the time interval 关 t,t⫹⌬t 兴 . By dividing this ⫺1 , ⌬N m (t) can be written as a sum of interval into smaller intervals 关 ␶ n , ␶ n⫹1 兴 of length ␥ m ⫺1 ␥ m ⌬t⫽O(␧ ) independent random variables N m ( 关 ␶ n , ␶ n⫹1 兴 ) 共number of jumps between ␶ n and ␶ n⫹1 兲, which have mean and variance 1. Therefore, by the central limit theorem, ⌬z n ⫽

⌬N m ⫺ ␥ m ⌬t

共82兲

冑␥ m

can be approximated for small ␧ by a Gaussian random variable of zero mean and variance ⌬t 关the convergence as ␧→0 actually holds for a fixed ␧-independent ⌬t; for ⌬t given by 共80兲, one gets an infinitesimal increment dz n 兴. Shifting t by ⌬t or changing m leads to independent increments ⌬N m and ⌬z m . It follows that ⌬z m are the infinitesimal increments of some independent real Wiener processes z m in the limit ␧→0. Indeed, a Wiener process z is by definition a stochastic process with independent increments ⌬z⫽z(t⫹⌬t)⫺z(t) distributed according to a Gaussian law of variance ⌬t. The convergence of (N m ⫺ ␥ m t)/ 冑␥ m to a Wiener process can be shown more rigorously by using a theorem proven in Ref. 32. The next step consists in evaluating both the mean and the fluctuating parts of 兩⌬␸典 to leading order in ␧. Let p⫽ 兺 m ⌬N m (t) be the total number of jumps between t and t⫹⌬t. We denote by s q ⫽t q⫹1 ⫺t q , q⫽1,...,p⫺1, the time delays between consecutive jumps and set s 0 ⫽t 1 ⫺t and s p ⫽t⫹⌬t⫺t p . By 共79兲 and 共80兲, p and s q have mean values ␥ ⌬t and ␥ ⫺1 of order ␧ ⫺1 and ␧ 4 , respectively. The generalized Hamiltonian 共68兲 can be decomposed into two parts: K⫽K 0 ⫺i␧ ⫺2 R,

共83兲

where K 0 and R are ␧-independent: K 0⫽

1 2i

兺m L m† L m ,

R⫽

兺m 冑¯␥ m L m .

共84兲

⫺1/2 L m q if q⫽1,...,p and V 0 ⫽0. With the help of 共69兲, 共81兲, and 共83兲, one obtains Let us set V q ⫽ ¯␥ m q

再兿冋冉 0

兩⌬␸典⫽

q⫽p

冊

册冎

1 1⫺␧ ⫺2 s q R⫹ ␧ ⫺4 s 2q R 2 ⫺is q 共 H⫹K 0 兲 ⫹O共 ␧ 6 兲共 1⫹␧ 2 V q 兲 ⫺1 兩 ␸ 共 t 兲典 , 2 共85兲

where the product is taken in decreasing order in q. The terms of order ␧ in the expansion of the product are p

兩⌬ ␸ 典 ( f )⫽

⫽

兺

q⫽0

共 ⫺␧ ⫺2 s q R⫹␧ 2 V q 兲兩 ␸ 共 t 兲典

兺m 共 ⫺␧ ⫺2 ⌬t ¯␥ m1/2⫹␧ 2 ⌬N m共 t 兲¯␥ m⫺1/2兲 L m兩 ␸ 共 t 兲典 ⫽ 兺m ⌬z m共 t 兲 L m兩 ␸ 共 t 兲典 .

共86兲

These are the leading order fluctuating forces 共of zero mean兲. Since fluctuating terms of higher order should not contribute in the limit ␧→0, we may replace the product in 共85兲 by its mean value in computing the terms of order ␧ 2 and more. This simplifies greatly the computation, because the order of the operators in the product then becomes of no importance. In fact, by the remark following 共71兲, the random variables s 0 ,...,s p , m 1 ,...,m p are independent, so that the mean of the ⫺1/2 L m ⫽R/ ¯␥ . Therefore, one product is the product of the means. Moreover, MV q ⫽ 兺 m ( ␥ m / ␥ ) ¯␥ m


3534



can commute all operators in the product when computing the mean value. It is easy to show that the terms of order ␧ 2 cancel on average. The terms of order ␧ 3 are found to be proportional to ⌬t: 兩 ⌬ ␸ 典 (d) ⫽M兩 ⌬ ␸ 典 ⫽⫺i 共 H⫹K 0 兲 ⌬t 共 1⫹O共 ␧ 兲兲 .

共87兲

This is the drift contribution to 兩⌬␸典. The previous computation shows that the QJ dynamics is transformed as ␧→0, if one looks at it with the time resolution ⌬t⫽ ¯␥ ⫺1 ␧ 3 , to a diffusive dynamics given by the linear Itoˆ stochastic differential equation:12,15,17

冋

兩 d␸ 典 ⫽ ⫺i 共 H⫹K 0 兲 dt⫹

册

兺m L m dz m共 t 兲兩 ␸ 共 t 兲典 .

共88兲

† L m /2i and dz m (t) are the infinitesimal increments of independent real Wiener Here K 0 ⫽ 兺 m L m processes z m , which have zero mean and satisfy the Itoˆ rules:

dz m 共 t 兲 dz n 共 t 兲 ⫽ ␦ n,m dt,

共89兲

dz m 共 t 兲 dt⫽0.

D. Nonlinear quantum state diffusion

We have so far determined the linear stochastic Schro¨dinger equation for 兩 ␸ (t) 典 . The corresponding nonlinear equation for the normalized wave function 兩 ␺ (t) 典 is obtained by means of the above-mentioned change of probability P→ P ⬘ . This derivation of the nonlinear QSD equation from the linear one is actually well-known.12,28 It is slightly more complicated than for the QJ dynamics, because 兩 d␺ 典 is to be expressed in terms of Wiener processes for the new probability P ⬘ . This can be done by using Girsanov’s theorem,33 which states that a Wiener differential dw m for P ⬘ is obtained by adding an appropriate drift differential to dz m . For the change of probability defined by 共73兲, the conditional 共mean兲 expectation of dP ⬘ /dP given 兩 ␸ (t) 典 is 储 ␸ (t) 储 2 . The drift differential is then ⫺ 储 ␸ (t) 储 ⫺2 d储 ␸ 储 2 dz m 共for more details, see Refs. 12, 28兲. From 共88兲 and 共89兲, one gets d储 ␸ 储 2 ⫽2 储 ␸ 共 t 兲储 2

兺m ᑬ具 L m 典 t dz m ,

共90兲

which implies that 储 ␸ (t) 储 2 is a local martingale. Thus, dw m 共 t 兲 ⫽dz m 共 t 兲 ⫺2ᑬ具 L m 典 t dt.

共91兲

According to Itoˆ’s formula,30 one has

冉

冊

兩 d␸ 典 d共冑储 ␸ 储 2 兩 ␺ 典 ) d 储 ␸ 储 2 d 储 ␸ 储 2 d 储 ␸ 储 2 兩 ␺ 典 ⫹ 兩 d␺ 典 ⫽ ⫽ 兩 d␺ 典 ⫹ ⫺ . 储␸储储␸储储␸储 2储␸储 8储␸储3

The multiplication of both members by dz m leads, with the help of 共88兲–共90兲, to dz m 兩 d␺ 典 ⫽(L m ⫺R具 L m 典 t ) 兩 ␺ 典 dt. Going back to the original equation, it follows:

冋

兩 d␺ 典 ⫽ ⫺i 共 H⫹K 0 兲 dt⫹

兺m ᑬ具 L m 典 t

冉

冊

3 ⫺L m ⫹ ᑬ具 L m 典 t dt⫹ 2

册

兺m 共 L m ⫺ᑬ具 L m 典 t 兲 dz m 兩 ␺ 典 .

The nonlinear QDS equation is obtained by replacing 共91兲 into this equation:11,12,15

冋

兩 d␺ 典 ⫽ ⫺i 共 H⫹K 0 兲 dt⫹

兺m ᑬ具 L m 典 t

冉

冊

1 L m ⫺ ᑬ具 L m 典 t dt⫹ 2

兺m 共 L m ⫺ᑬ具 L m 典 t 兲 dw m共 t 兲

册

兩␺共 t 兲典.

共92兲




3535

FIG. 10. The links between the different stochastic Schro¨dinger equations.

Let us come back to the mode dynamics of Sec. III. If one sticks to the norm-preserving QJ dynamics, the jump operators W ⫾ in 共31兲 are defined up to a multiplicative constant. If k⫽0, ã ⯝a, they can be obtained, up to such a constant, from the Lindblad operators L ⫹ ⫽ 冑¯␥ ⫹ a † and L ⫺ ⫽ 冑¯␥ ⫺ a by means of formula 共66兲, provided that the transition rates are replaced by some effective rates ␥ ⫾ ⫽ ¯␥ ⫾ 兩 ⑀ 兩 2 and L ⫾ are multiplied by the phase factors e ⫾i ␪ , ␪ ⫽arg ⑀. Thus the above-presented analysis can be used to compute the QSD equation for the normalized mode wave function 兩 ␺ (t) 典 in the limit of large laser fields 兩⑀兩→⬁. The introduction of the phase factors e ⫾i ␪ in 共92兲 leads to the QSD equation 共54兲. E. Links between the stochastic Schro¨dinger equations

The summary of the results of this section is given in Fig. 10. The stochastic Schro¨dinger equations in Itoˆ form for the linear and nonlinear QJ models are, respectively, Eqs. 共65兲 and 共78兲, and those for the linear and nonlinear QSD models are, respectively, Eqs. 共88兲 and 共92兲. VI. CONCLUSION

We have shown that the nonlinear quantum jump 共or Monte Carlo wave function兲 model applied to a simple optical system 共damped harmonic oscillator at finite temperature T兲 can be generalized to describe the evolution of the quantum field in a cavity monitored by an atomic beam of two level atoms. These atoms cross one by one the cavity and interact at its exit first with a classical laser field E, and then with a detector measuring their states. This kind of monitoring by measurements is similar to that obtained by homodyne measurement of the field in a decaying cavity,5,8 the photon counting on the output field being replaced by the measurements on the atoms. Actually, if all atoms are sent in their ground state (T⫽0), the stochastic evolution of the wave function of the quantum field 共quantum trajectories兲 is the same for the two kinds of monitoring. If the atoms form a beam of randomly prepared atoms with temperature T⬎0, they may also emit photons in the cavity and thus a new kind of quantum jump comes into play 共creation of a photon兲. This has notable effects on the quantum trajectories. The effect of the laser field E is to modify the two jump operators. In fact, the measured atomic transitions can be driven by this field as well as by the interaction with the studied quantum field in the cavity. As a result, E also modifies the generalized Hamiltonian that rules the evolution between jumps. The average over all realizations of the measurements leads to an E-independent dynamics, described by a density matrix satisfying the master equation of the damped harmonic oscillator with temperature T. Whereas the density matrix converges at large times to the Bose–Einstein equilibrium, the individual quantum trajectories for given realizations experience localization toward squeezed states at large E. The squeezing parameters evolve to some almost constant values, up to small fluctuations going to zero in the infinite laser intensity limit. This localization occurs at large enough times, for any initial state of the quantum field. The centers of the squeezed states move


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randomly in the complex plane, in such a way that the time averages of the quantum probabilities to find n-photons are distributed according to Bose–Einstein 共ergodicity兲. The squeezing amplitude r and phase ␾ are controlled, respectively, by the temperature T and the laser field E. r is found to increase with T, which means that the squeezing is enhanced by increasing the temperature of the atomic beam; however, the waiting time before r reaches its almost stationary value is also temperature increasing. On the other hand, no squeezing is obtained at T⫽0, and localization toward Fock states occurs if E⫽0. As in the case of the homodyne measurement, the quantum trajectories are given in the infinite E limit by a so-called quantum state diffusion 共QSD兲 stochastic Schro¨dinger equation, involving real white noise.5,8,11,12 A precise mathematical derivation of this equation from the quantum jump dynamics was performed in Sec. V for arbitrary open quantum systems having a Lindblad-type dynamics. More precisely, this derivation starts from a linear version of the QJ dynamics proposed in Ref. 18, in which the wave function is not normalized at each step, which is proven to be related to the nonlinear QJ model by a simple change of probability. ACKNOWLEDGMENTS

D.S. is grateful to Rolando Rebolledo and Walter Strunz for fruitful discussions. M.O. acknowledges the financial support of the Project Fondecyt 1010777. D.S. acknowledges the financial support of Fondecyt by Postdoctoral Grant No. 3000035 and the ECOS program of scientific exchanges between France and Chile. 1

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