Stochastic Quantum Dynamics of a Continuously Monitored Laser

PHYSICAL REVIEW A VOLUME 47, NUMBER 6 JUNE 1993

Stochastic quantum dynamics of a continuously monitored laser

H. M. Wiseman Department of Physics, University of Queensland, Queensland )072, Australia (Received 30 November 1992) The stochastic dynamics of the quantum state of a laser conditioned on continuous intensity and phase-sensitive measurements on its output are examined. We first develop the essential model for a Poissonian laser, and then generalize it for regularly pumped lasers. We show that the rate of phase diffusion (which gives the laser linewidth) is not affected by the regularity of the pump. In both cases, heterodyne detection causes the phase variance in the conditioned laser state to become very small, yet remain significantly above (at least three times) that of a coherent state. The phase difFusion is manifested by a random walk undertaken by the mean phase, with the stochasticity arising from the local-oscillator shot noise. In contrast, intensity measurements have no effect on a Poissonian laser. For a perfectly regularly pumped laser, coarse-grained intensity measurements (for which an approximate theory is developed here) collapse the state to one with an arbitrarily small photon-number variance-to-mean ratio,

PACS number(s): 42.50.Lc, 03.65.Bz, 42.55.—f, 42.50.Dv

I. INTRODUCTION domly. The source of the stochasticity is ultimately the shot noise in the heterodyne photocurrent which condi- The emergence of quantum optics as an important field tions the laser state. The magnitude of the conditioned of theoretical physics was precipitated by two important phase variance (three times that of a coherent state under technological developments: the laser and the efBcient ideal conditions) is a new and unanticipated result. photodetector [1]. Laser light is unlike other optical light More recently in quantum optics, great interest has sources in that it has a large coherent amplitude. To arisen in so-called squeezed states [7]. These are states treat this quantum mechanically, it was found extremely in which the quantum uncertainty in one variable (such as useful to use coherent-state representations, such as the one quadrature or the photon number) is less than that Glauber-Sudarshan P function [2,3]. Also, it was nec- of a coherent state, and that in the conjugate variable essary to develop new techniques for dealing with open (such as the other quadrature or the phase) is greater quantum systems, such as the master equation [4]. Both than that of a coherent state, as necessitated by Heisen- of these methods are related to photodetection. Coher- berg's principle. There has been considerable theoretical ent states have been found to be precisely those states work on laser systems which have photon-number vari- which are not affected by conventional detection tech- ances below the classical (coherent) limit, and so produce niques. The relationship between the openness of quan- sub-Poissonian photon statistics [8—16]. Here we consider tum systems and their continuous monitoring is still be- ideal lasers with a sub-Poissonian pump, and derive a ing explored. simple master equation for such lasers. This yields a In this paper we examine many of the early themes in number of results. Firstly we show that the linewidth, quantum optics using very recently developed theories of and the random phase walk under heterodyne detection, quantum measurement [5,6]. While not leading directly is the same as in Poissonian lasers. Now, however, the to new experimental predictions (although see the final conditioned state is a (close to minimum uncertainty) paragraph of Sec. VIII), these techniques provide new squeezed state, rather than a mixture of coherent states insights into the stochastic dynamics of individual quan- as in a Poissonian laser. tum systems. In particular, we clarify the distinction be- Unlike the Poissonian laser, regularly pumped lasers tween coherent states and the quantum state of the light are affected by intensity measurements. Direct photode- inside a single mode laser. To do this, we first develop a tection is shown to collapse the conditioned state of the simple master equation for an ideal laser producing light cavity mode into one with a photon-number variance with Poisson photon statistics, as in a coherent state. An below that of its steady-state unconditioned operation. immediate consequence of this master equation is that The extra variance in the unconditioned (without mea- the phase of the laser diffuses, with a linewidth inversely surement) state comes from classical random variation proportional to the mean photon number. An initially in the conditioned mean intensity. As in the random- coherent state inside the cavity becomes a mixture of phase walk, the source of this randomness is the shot coherent states over all phases. The picture of a laser noise in the measured photocurrent. To derive this re- state with a well-defi. ned phase is restored if we incor- sult, it was necessary to develop an approximate theory of porate heterodyne detection. Using a stochastic master direct photodetection in which the state is conditioned by equation recently derived for such measurements [6], we the photocurrent. Strictly, direct photodetection should show that the laser state is collapsed by a measurement be treated in terms of individual photodetections, but by to a near-coherent state with a phase which wanders ran- coarse graining in time we heurestically derive a plausi-

1050-2947/93/47{6)/5180(13)/$06. 00 47 5180 1993 The American Physical Society 47 STOCHASTIC QUANTUM DYNAMICS OF A CONTINUOUSLY. . . 5181

ble stochastic master equation which is valid for a suit- scale the eIFect by ¹ The density operator W for the able class of systems. The theory shows that the condi- field plus one atom obeys the following master equation: tioned photon-number variance to mean ratio in a regularly pumped laser can be arbitrarily small in principle. W = —iy[a o i2 + ao i2, W] + p D[oer] W + P 'D[o op]W The organization of this paper is as follows. Firstly, +r. D[a]W. (2.1) in Sec. II we develop the essential model of a Poissonian pumped laser, and derive some simple results such as the Here, 'D[c] is a superoperator taking the arbitrary oper- laser linewidth. In Sec. III we review the quantum theory ator e as an argument, defined by of heterodyne detection which we derived elsewhere [6], and motivate the theory from simple considerations. This D[c] = &[c] —A[c], (2.2) theory is applied to the Poissonian laser in Sec. IV, and a physical picture of a laser with a randomly wandering where we are using the following notation: phase is rigorously defined in terms of conditioning the = state on the heterodyne photocurrent. The quantum the- g[c]p cpct, (2 3) ory of the ideal laser is generalized in Sec. V to lasers with A[c]p = 2r(ctcp+ pctc), (2.4) a non-Poissonian pump. In Sec. VI we present an approximate quantum theory of direct photodetection, which is where p is an arbitrary density operator. valid for systems such as a laser at steady state. The To achieve amplification by stimulated emission, it is effect of such detection on the generalized laser model is necessary for level ll) to be rapidly depleted so that a examined in Sec. VII, as well as the efFect of heterodyne population inversion on the lasing transition occurs. This detection. Section VIII is a discussion. requires that the spontaneous-emission rate p be large, in the following sense:

2 II. ESSENTIAL MODEL OF A I ASER XP &(1 (2.5)

section we In this present the simplest possible phys- where is the mean photon number for which an ex- ical model for a laser. reproduces the essential fea- p It pression will be given later. However, this condition is The num- tures of an ideal laser. steady-state photon not sufIicient to produce a laser. We also require that ber distribution is Poissonian, as in a coherent state, the stimulated emission events be Poissonian distributed but the phase difFuses, giving a finite linewidth. The in time in order to produce a Poissonian photon-number laser consists of single-mode optical cavity containing a distribution. This will occur if the following condition is atomic-level is shown N three-level atoms. The structure satisfied: in Fig. 1. The lowering operator from an upper to a lower level is denoted o~„. The cavity mode has the annihila- (2.6) tion operator a, freely rotating at frequency ~. This is resonant with an atomic transition between levels ll) and l2) . The dipole coupling constant is y, which we take to Finally, if we wish to adiabatically eliminate the atoms, be real for simplicity. The lower level ll) spontaneously then the cavity field must relax much more slowly than decays to the atomic ground state l0) at rate p. The the atomic populations. This will be satisfied if atoms are incoherently pumped, causing the atom to be excited from l0) to l2) at rate P. Finally, the cavity mode (2 7) is damped to the external continuum of electromagnetic modes at rate r. Now eventually we will adiabatically These three conditions could be satisfied (with e 6 eliminate the atoms, so we need consider only the inter- 0.1, g 10 ) by a set of realistic parameters such as action between the field and one atom, and subsequently p" lQ s P Is p 10 s r, g 10 s p~10 I2) Now the above master equation (2.1) permits a solution of the form

w(t) = lo)&ol @p"(t)+ 12)&2I @p"(t) + 12)&IIp"(t)+H' +II)&IIp"(t) (2.8) 10) Here the p's are operators in the Hilbert space of the cavity mode. The leading term p is of order 1, while FIG. 1. Schematic diagram of atomic transitions in the the others are much smaller, due to the above scalings. ideal laser model. The dipole coupling constant (one-photon Specifically, Rabi frequency) for the ll) —l2) lasing transition is y. Level l0) is incoherently excited to level l2) at rate P. The sponta- "= o(q), "= o("~')o(q), "= o(.)o(q). neous-emissiou rate of ll) to l0) is p, and that of level l2) is assumed negligible. (2 9) 5182 H. M. WISEMAN 47

These scalings will be shown to be self-consistent. The the following master equation for p: four cavity-mode p operators obey the following coupled equations: p = &P [a'l((&[a'9 'P) —P] + ~D[a]p (2») (2.10a) Multiplying the pumping term by the number of atoms P22 P12 P21 00 22 N, we find the following elegant form for the master equa- &+( t) + p + ~[ ] (2.10b) tion of an ideal Poissonian laser: 21 ll 21 21 &+(a P22a) + ~[ ) (2.10c) 2 rc 'p = (@8[at] 'D[a]) (2.20) ll t 21 12 ll ll + p, ( ) + ~[ ) (2.10d) where 8 is defined by Consider first Eq. (2.10c). Using the above scalings (2.5)—(2.9), this can be rewritten as F[c] = J'[c]A[c] ' —1. (2.21)

[1+&(~) + O(~)O(~)]P" = imp"a - 2P", (2.11) This is to be compared with the definition of 'D (2.2). These simple definitions show clearly the distinction be- where the terms on the right-hand side are of equal order. tween the Poissonian damping process 'D[a] with rate r Now this shows that p relaxes at a rate of order p, times the mean photon number, and the Poissonian ex- whereas p will be shown to relax at a rate of order ~ citation process t [at] with fixed rate K times p where slower than this. Thus it is valid to say that p will be = slaved to p, and replace the former by the steady-state p, Wp/K. (2.22) value. Ignoring lower-order terms in Eq. (2.11) gives Using the Fock basis for the cavity mode gives the following: p21 +p22aQ(~1/2)Q(p22) (2.12) y (2+nm — @=Pl , P &, & P, )— Substituting this into Eq. (2.10d) gives the following: n+m 4&' +g(n + 1)(m + 1) p„+1 ~+1 — (n + m) p„~. [1+O(b)]p" = a p a —pp". (2.13) 2 y (2.23)

As above, p is constant on the time scale over which Here, as in the remainder of this paper, we are measuring p relaxes to its steady-state value of time in inverse units of the cavity linewidth v. It is easy to see that p, is the steady-state mean photon number by 4 2 &[a']p" = &(e)&(p") (2.14) verifying that the steady-state density matrix satisfying , Eq. (2.23) is where g is as defined in Eq. (2.3). Now we substitute ~p Eq. (2.12) into Eq. (2.10b) to get p„(oo) = 6„, e (2.24) 4 —4X This photon-number distri'~ution is Poissonian, as in a [1+O(~))p" = &[a')p" + &p", (2.15) coherent state of amplitude Unlike a coherent state, y i/p. it has a completely undefined phase as the ofF-diagonal where A is as defined in Eq. (2.4). The relaxation time elements are zero. This can be seen from an alternate of p is of order 1/b times that of p0 so once again it is representation of the steady-state density operator using possible to replace the former by its slaved value, the Glauber-Sudarshan P function,

p" = (&[a') p" = &(() (2.16) = (2.25) 4, ) P(~) 2 Substituting this into gives Eq. (2.14) Thus, the stationary density operator can be expressed either as a Poissonian mixture of number states (with (2.17) maximally determined intensity and completely undefined phase), or as a mixture of fixed amplitude coherent Now the reduced density operator for the cavity mode states (with well-defined intensity and phase). Why then is given by is a laser often treated as being in a coherent state of unknown phase, but never as being in an imprecisely P=Tr~« ~=P00 +P11 +P22 (2.18) known Fock state? (Here we are talking of classical knowledge, in that the imprecision is merely due to the Since the latter two terms are much smaller than the experimenter's ignorance. ) first (by a factor of order ( 10 s with the parameters The basic answer to the above question is differential suggested above), we can ignore these and approximate lifetimes, as pointed out by Gea-Banacloche [17]. If a p by p00. Substituting Eq. (2.17) into Eq. (2.10a) gives laser were to be put into a Fock state, then under the 47 STOCHASTIC QUANTUM DYNAMICS OF A CONTINUOUSLY. . . 5183 master equation (2.23), the probability of it remaining in III. QUADRATURE MEASUREMENTS that state will decay at a rate of order Kp, . Using the parameters suggested above, the lifetime of a number state The fact that the inverse of the laser linewidth is of is thus of the order of 10 s. On the other hand, the order minutes is often used as a justification for approx- survival probability of a coherent state of amplitude ~p, imating the laser by a coherent state. On the time scale decays at a rate of order r/p, giving a lifetime of order of many experiments, a laser initially in a coherent state minutes. This can be seen by calculating the infinites- will remain in that state to a very good approximation. imally evolved density matrix which is initially in the However, this appears to beg the question of how the ob- coherent state ~o.) with ~a;~ = p. . From (2.23), the result laser got into a coherent state in the first place. An is vious answer is that measuring the phase of the laser "a* — collapses it into a coherent state (or alternatively, deter- !2 o. t dt(n m)2& = e ! 1 — . mines which coherent state it really was in). In practice, ~ p„~(dt) ~ (226) n!m! 2 n+m phase measurements are made using homodyne or heterodyne detection. In this section we review briefly the For large p, , it is an excellent approximation to replace quantum theory of such phase measurements which we m in above denominator n+ the by 2p, . Then, the above have presented in detail elsewhere [5,6]. short-time result (2.26) could have been obtained from To investigate the collapse of the laser state due to the following master equation: measurement we need a theory which describes the Se- tective evolution of the system. That is to say, we need = I"D[o,ta]p, p (2.27) to know what the state of the laser is given the result of the measurement. The nonselective evolution of laser where we have defined (in units of r) the is simply given by the master equation (2.27). Different I = 1/2p. (2.2S) measurement schemes on the output light of the laser will give different selective evolution equations. If the result This short-time master equation is valid for an initial of the measurement is averaged over, these selective evo- coherent state of amplitude ~p. By linearity, it will also lutions necessarily reproduce the nonselective evolution be valid for all initial states which are mixtures of such for the density operator. coherent states. But it is easy to verify that the class C„ The simplest kind of measurement on the output light of such mixtures is invariant under the master equation is direct photodetection. The selective evolution of the (2.27). Thus, this approximate master equation is in fact system is easy to describe. In the infinitesimal time in- valid for all times. terval (t, t + dt), a photodetector of quantum efficiency From we can derive the following result: (2.27), g placed at the output of a cavity will register a count with probability qP, (t)dt, where P, (t) = Tr(+[a]p, (t)}. d I' (2.29) Recall that we are measuring time in inverse units of the cavity linewidth. The subscript c on p, (t) indicates that the state of the system at time t is in general conditioned Then, by the quantum regression theorem and the fact on previous photocounts. If a photodetection occurs, the that is a member of if is a member, we have ap C„ p new state of the system is p, (t +dt) = +[a]p,(t) /P, (t). If no photodetection occurs, the state of the system evolves ( (&) (o)) = P (2.30) such that the average evolution of the system is given by its master equation. This is shown explicitly in Ref. [6]. Thus the laser has a Lorentzian spectrum with full width It is possible to generalize this theory of photodetec- at half maximum equal to I'. This is as expected since tion by adding a local oscillator to the amplitude of the a master equation containing a double commutator with output field before it is detected. This is achieved by a!'a will cause diffusion in the conjugate variable phase. using a beam splitter, and it is advantageous to use two This is perhaps more easily seen from the Fokker-Planck detectors, one for each of the beams leaving the beam equation for the P function. Since this is nonzero only for splitter. If the local oscillator is tuned to cavity reso- complex amplitudes n of modulus ~p, the P function has nance, then we have a balanced homodyne measurement only one real argument, p = arg(n). It satisfies a pure of the quadrature of the system in phase with the local diffusion equation oscillator. For monitoring both quadratures (or equiva- lently, the amplitude and phase) of the system, one can (2.31) use so-called eight-port homodyne detection, where the output light from the cavity is split into two independent balanced homodyne detectors. The same result can The Green's function for this equation [4] be achieved by using heterodyne detection, in which the local oscillator is significantly detuned from the cavity. (2.32) The two Fourier amplitudes of the resulting photocurrent at the detuning frequency give the two quadrature measurements. is an alternate route to the linewidth stated above, which Since we are interested in monitoring the phase of the agrees with that of Louisell in the same limits [4]. laser, this last scheme is most appropriate. We measure 5184 H. M. WISEMAN the phase of the laser relative to that of the local oscilla- where we are using the notation E[A, B] = E[AB]— tor, which is defined as a coherent state with amplitude E[A]E[B]. From standard quantum regression tech- P. If the local-oscillator amplitude is much greater than niques [19],this is given by the system amplitude, then the rate of photocounts is much higher than it would be without the local oscilla- ~(t, t + z) = rl Tr(ai e [ap(t)] j but the in the conditioned system state due tor, change — ' to each photodetection is much less. In the limit where rl'T [ P(t)]T [ P(t)]+nti( ) (36) the local-oscillator amplitude goes to infinity, it is possible to convert from discrete photocounts to a continuous where p(t) is the density operator at time t Th. at this photocurrent. Denote the photocurrent measuring the is assumed given is the reason that the current at time quadrature in phase with the local oscillator by '(t), t in Eq. (3.5) is not conditioned; whether or not the I; mea- and that out of phase by I,""(t). Then it is useful to de- system state at time t is conditioned on previous fine a "complex photocurrent" I;"r (t) = I; '(t)+iI "(t), surement results is irrelevant. The current at time t + ~ which is given simply by is, however, conditioned on the results of the homodyne measurements in the interval [t, t + ~). In particular, the (3.1) state of the system at time t + ~ is conditioned on the noise in the photocurrent at time t. - where Gaussian white noise sat- ((t) represents complex It is easy to see [5) that, upon substituting Eq. (3.1) isfying into Eq. (3.5) and using Eq. (3.2), the terms remaining after cancellations are E(((t)( (t )) = ~(t —t ) (3.2) t = E(Tr [ala'p, (t q6'(~). (3. with all other first- and second-order expectation values R(t, + ~) + ~)]~rl(*(t)) + 7) vanishing. Such ensemble averages denoted are not by E For this to agree with Eq. (3.6), we simply require that to be confused with conditioned quantum averages such as (ai),(t) which is given by Tr[atp, (t)]. The condition- = — E( (t+ )( (t)) v ([ (t)] T [ (t)] (t)). ing of the system density operator on the photocurrent . is given explicitly via the following equation (which is to (3.8) be interpreted in the Ito formalism of stochastic calculus [»]): The left-hand side of this equation is not zero because the conditioned density operator at time t + ~ is influenced P.(t) = &P.(t) + Wnf((t) [a —(a) (t)] + H e )P.(t) (3 3) by the photocurrent noise at time t. Specifically, we see from our postulated Eq. (3.4) that Although Eq. (3.3) is generally unfamiliar in content dt)(*(t) = and in form, we can motivate it from three simple prin- ,(t+ ~E .(t), (3.9) ciples. the conditioned evolution must Firstly, system where we have used Eq. (3.2) and discarded terms of be completely determined from the conditioned complex higher order in dt. Between t+ dt and t+7., the ensemble photocurrent This is true the definition of a con- (3.1). by average evolution of p, (t) is simply given by the Liouville ditioned system state. Secondly, if we ignore the result of superoperator l:. That is, we get to lowest order in dt the measurement by averaging over all possible results, we must return to the original master equation for the E(p, (t+ ~)(*(t)) = v ale Pp(t)]. (3.10) unconditioned state j = Zp. In this case, this amounts to setting the noise terms in (3.1) equal to their average A more rigorous derivation of this result is given in Ref. value of zero, or to setting the efIiciency of the detector [5]. Comparison with Eq. (3.8) shows that we must define to zero. Thus, the only additional terms in the master the superoperator T by equation must be linear in the photocurrent noise. Since the stochastic evolution equation must preserve the Her- Pp = (a —Tr[ap])p. (3.11) miticity of p, (t), the obvious solution suggested by these first two principles is This is precisely the definition needed to reproduce the correct equation (3.3). (t) = & (t) + ~[((t)& (t)+ H (3.4) By use of this equation, we will show that the condi- . . . tioned laser state does have a well-defined phase at all where T is an as-yet undetermined superoperator which times, relative to the fixed phase of the local oscillator. may be nonlinear in its action on p, (t). However, this appears to beg the question as to how the The third principle is that the stoehastie equation (3.4) fixed phase of the local oscillator is produced. In fact, it is must yield the correct correlation functions for the com- not necessary for the local oscillator to have a fixed phase; plex photocurrent (3.1), as found by more standard anal- it can come from a completely phase-dift'used laser. Pro- yses. For example, the two-time correlation function for vided that the local-oscillator linewidth is much less than the heterodyne photocurrent is defined as the ensemble that of the system, the dynamics of the relative phase average will be dominated by the system dynamics, as given in Eq. (3.3). The absolute phase is as undetermined as that R(t, t+~) = E(I;" (t+~), I (t) ), (3.5) of the local oscillator. This is usually unimportant, be- 47 STOCHASTIC QUANTUM DYNAMICS OF A CONTINUOUSLY. . . 5185

cause in practice only relative phase is measurable at less) than that of a coherent state, then direct photode- optical frequencies. tection would have an effect on the conditioned system state. Moreover, the effect is distinctly different in the IV'. MDNITDRINC THE POISSONIAN LASER. two cases, as will be seen in Sec. VII. Heterodyne detection, on the other hand, has a dra- matic effect on the laser at steady state. The stochastic First we determine the conditioned state of the laser master equation for the conditioned laser state at steady under direct photodetection. We showed in Sec. that II state is the laser at steady state is in a mixture of coherent states of amplitude of modulus ~p, . From the theory in Sec. III, P, (t) = rV[~ta]p it is easy to see that direct photodetection has no efFect — on the laser state. This is because the coherent states +V &(&(t)[&p.(t) (&).(t)p.(t)] + H c ). (4.1) ~o.) are eigenstates of the jump superoperator J'[a], and the eigenvalues are ~n~, which is independent of phase. As noted in Sec. II, the P function for the laser at steady If the uncertainty in the laser intensity was greater (or state is a function of the phase y only. It obeys

I' Bs = — e*t' —, dp'e'" . P, (p, t) ~ + +gyes(t) P, (p't) + c.c, P, ((p, t). (4.2) g2 Olp ))

Now we will show shortly that the long-time solutions U = QI'/2r) p. (4.10) to this equation have a variance in &p of order 1/p (( 1. Keeping terms of lowest order in 1jp in Eq. (4.2) gives If there is no excess phase diffusion in the laser, then 1 = 1/2p, as derived ln Sec. II. Assuming ln addition that I )9 — g = 1, this gives the minimum steady-state conditioned &) = + ~) Ph, —,v'2nv(hatt))v 4.(~)l) P.h~, P phase variance of U = 1/2p, . As the mean 2 B(p photon number p, goes to infinity, this goes to zero. Nevertheless, (4.3) on a quantum scale, it is always significantly higher than Here, (~(t) represents real normalized Gaussian white the value of zero for a coherent state. In fact, it is equal to noise, defined by the Q-function phase variance of a coherent state. Thus it is not true that monitoring the phase of a laser collapses g~(t) = -imp~"~-f'lg(t)] (4.4) it to a coherent state. Equation (4.9) shows that the phase variance relaxes and is the central angle of the distribution P, (t) to its steady-state value at a rate of order the cavity linewidth, which is of order p times the laser linewidth. 4.(t) = dV P.(V»t)V' (4 5) Thus, we may replace U, (t) in Eq. (4.7) by its steady- state value. This gives Since the variance in the phase is very small, we can ignore the periodicity requirement on P, ()p, t) and instead (4.11) use the following ansatz for the solution of Eq. (4.3): This equation is precisely of the form of the stochastic — differential equation for the laser phase which would be P.(V» t) = exp(- lV &.(t)]'/2U. (t)) (4.6) /2~U, (t) derived from the original Fokker-Planck equation for the P function (2.31). That is not to say, however, that all If P,(t) is initially in such a Gaussian state, it will remain we have achieved is a very lengthy and obscure deriva- so under the evolution of Eq. (4.3), with the conditioned tion of a standard result. The phase P, (t) is not a math- mean and variance evolving via ematical artifact. It has a physical interpretation as the = mean phase of the laser, conditioned on the results of the 4.(t) v'2n~&. (t)U.(t), (4.7) continuous heterodyne measurement. The conditioning U, (t) = I' —2gp U, (t) . (4.8) occurs via the noise in Eq. (4.11) which is physically derived from the photocurrent shot noise, not merely a These equations are derived in Appendix A. Note that formal device producing a stochastic equation equivalent the effect of the measurement on the variance is deter- to a Fokker-Planck equation. We thus have the follow- ministic, and causes it to reduce as expected. If the laser ing picture for the dynamics of a continuously monitored state is taken to be initially in a coherent state, the nor- standard laser. The laser state is a Gaussian mixture mally ordered variance is U, (0) = 0. From Eq. (4.8) we of equal-amplitude coherent states with a constant phase then see that variance inversely proportional to the mean photon num- U, (t) = U tanh(2r)I'pt), (4.9) ber. The mean phase of the mixture undergoes a random walk on a time scale inversely proportional to the laser where the steady-state P-function phase variance is linewidth. 5186 H. M. WISEMAN 47

V. REGULARLY PUMPED LASER MODEL than 1. If the mean rate of excitations is p, as in Sec. II, then we ca,n write The laser model developed in Sec. II was based on the assumption that the excitation process was Poissonian. n(dt) = ddt + Qp, (q+ l)dW, (5.1) Specifically, this was achieved by assuming that the rate of decay from the lower lasing level to the ground state where de is an infinitesimal Weiner increment and q is the Mandel Q parameter [20] for the excitation process, was much greater than the rate of excitation from the — ground state to the upper lasing level. This assumption equal to 0 for a Poisson process and 1 for a perfectly las- has the benefit that the Poissonian laser master equation regular process. For an m level (including the two levels) atomic with matched transition has a simple form, and that the laser steady state is a ing system rates, = — — — Now the efFect of one stim- mixture of equal-amplitude coherent states, which gives q (m 2)/(m 1) [14]. Poissonian photon statistics in the output light. Other ulated emission is, from. Sec. II, given by the excitation superoperator 8[at]. Thus, the infinitesimally evolved assumptions regarding the decay and excitation rates in a density operator is three-level atomic system can give rise to sub-Poissonian statistics [12]. Furthermore, the more atomic levels in- (5.2) volved, the better the quantum noise reduction in the laser output [13,14]. As the number of levels increases Since 8[at] represents a small change to the state of the indefinitely with matched transition rates equal to half mode (adding one photon to p photons), we can assume the it is pump rate, possible to get arbitrarily low pho- that it is much less than unity, and expand Eq. (5.2) to tocount variance-to-mean ratios for long times [14]. get The explanation for this quantum noise reduction is that the reexcitation of the upper laser level following a p(t + dt) = (1 + n(dt)f [ai] stimulated emission is regularized due to the many inter- +2n(dt) [n(dt) — [ai] (5 3) mediate steps. It is important that the transition rates l]f )p(t). are matched; if one is much slower than the others, then this becomes the rate-determining step, and the excita- Substituting in the above expression for n(dt) and aver- tion process becomes Poi.ssonian, as in Sec. II. A regular aging over the uncertainty in the number of stimulated electronic excitation can also be achieved by externally emissions gives the following master equation: imposing a regular pump process on the laser [15,16], such as by a regular injection of atoms. Either method p= p, Fc~ + —Sa~ p+Vap. (5.4) of sub-Poissonian excitation can be treated simply using the superoperators derived in Sec. II. The following Here we have restored the damping term with rate 1. For argument is similar to that of Ref. [16]. the Poissonian pump (q = 0) this reduces to that derived Consider a short time dt in which the number of stim- in Sec. II (2.20), as expected. ulated emissions n(dt) is expected to be much greater To elucidate Eq. (5.4), we write it in the Fock basis,

( — — — 2qgnm(n 1)(m 1) 2(1 q) gnm 2 —q pn, m P' pn —2,m —2 + pn — m —1 ( + )( + 2) 1, P, — +g(n + 1)(m + 1)p„+i,m+1 2 (n + m) p„,m. (5.5)

Lookirig at photon-number populations P = p, gives This can obviously be sub-Poissonian, with a minimum = — — of p, /2 when q 1. Furthermore, Eq. (5.7) allows 2 Q the steady-state second-order correlation function to be P„=p P„2+" (1 ——q)P„ .2 2 calculated exactly as +(n + 1)P„+1—nP„. (5.6) (: n(t + w)n(t):) = p + pqe— (5.10) From this we can easily derive the following for the mean and variance of the photon number: This gives the following normalized noise spectrum for the laser intensity output: n=p —n, (5.7) — o2 = n+ (1+q) p, 2o. . (5.8) = S(~) p 1+ 1+(J (5.11) These equations show that the steady-state mean photon number is independent of q, as expected, and that the For perfect pump regularity, the noise is reduced to zero steady-state variance is given by at low frequencies. The above results for photon-number statistics are as ~' = V(1+ q/2). (5 9) obtained by previous workers. However, they make no 47 STOCHASTIC QUANTUM DYNAMICS OF A CONTINUOUSLY. . . 5187 mention of the effect of pump regularity on the laser function rather than the P function as in Sec. II be- phase. To investigate this, we convert the master equa- cause now we have to deal with nonclassical states (see, tion (5.4) into a Fokker-Planck equation for the Wigner for example, Ref. [19]). The result, which is derived in function W(o;, o."). It is necessary to use the Wigner Appendix 8, is

B — Cl 1 0 B2 B2 + +1+~ ~ ' +,. ~+ ~~+' ~~+ BA 2A 2 2 4ini2o. 2 Bn 4o. Bn' 4o. Bo.Bo." (5.12)

I This equation can be considerably simplified by trans- sities. Such a theory could be called a coarse-grained — in we must over forming to intensity and phase variables n = ~a~2 2 theory of photodetection, that average and y = ln(o. /n')/2i. Then the Fokker-Planck equation many individual photocounts in order to get a photocur- for W(n, Ip) becomes rent. However, this is not the approach we adopt here derive a stochastic equation for the conditioned sys- B to (n —p, ) tem state. Rather, we develop a heuristic model in the Bn manner of the motivation for the stochastic equation for 1 B 1B n+p heterodyne detection given in Sec. III. +-,[n+(1+~)/~]+-, B . „. Consider a cavity with a stable mean photon number n p, 1, and photon-number variance of order p. (5.1S) )) Let dt be a small-time increment such that the average From this, the results (5.7) and (5.8) may easily be veri- photocount in that time ( ddt) is very large, but the fied. change in the system ( dt) is very small. Obviously By replacing n by its mean p, in the diffusion terms, this requires p, )) 1, which will be satisfied in a laser we get the simple expression with p, 10 . Now since the change in the system over the time dt may be ignored, the photocount drn will be B 1 B2 1 1 B2 a Poissonian random variable with mean and variance ( —P)+ P( +&) — —— W. On 2+ 2 equal to gndt, where g is the efBciency of the photodetector. Since this is very large, it is well approximated by (5.14) a Gaussian random variable with equal first- and second- order moments. That is, we approximate dm by This clearly shows that regular pumping does not affect the phase-diffusion term. That is, reducing the inten- = sity fluctuations in this case does not increase the phase dm hundt + grlndW, (6.1) fluctuations. We thus expect the effect of heterodyne de- dR' tection on phase diffusion to be much the same as in Sec. where is an infinitesimal Weiner increment. Since IV. However, the effect on the laser intensity variance the mean photon number is assumed stable, we can re- needs to be investigated. Also, since we have a nonclas- place n by p, in the noise term of this photocurrent. This sical state, direct photodetection will now have an effect is necessary for the theory developed below. Then a pho- on the conditioned state of the laser. To deal with this, tocurrent defined by I(t) = dm(t)/dt will be given by we need the theory of coarse-grained intensity measurements developed in the next section. I.(t) = n(a'a). (t) + V'~p4(t) (6.2)

where ((t) = dW(t)/dt Here we hav.e added a condi- VI. CDARSE-CRAINED INTENSITY tioned subscript to the mean photon number (afa), (t) = MEASUREMENTS Tr[ap, (t)a ] in anticipation of the impending develop- ment of the theory. The quantum theory of direct photodetection has al- Now we wish to know how the photocurrent (6.2) con- ready been presented in Sec. III, in terms of the prob- ditions the system state. Presumably, it would be possi- ability of a photodetection occurring in an infinitesimal ble to derive this from the summation of the individual time increment, and the effect on the conditioned system effect of each photocount. However, this appears to be state. However, this theory is of limited use in prac- very diKcult, so instead we derive an evolution equation tice, because with many systems the photon lux is so for the conditioned state based on the three conditions high that only a photocurrent is measurable, not a pho- specified in Sec. III. These were (i) the stochasticity must tocount. From a theoretical point of view, the noise as- be determined by the measured photocurrent; (ii) averag- sociated with stochastic arrival times of photons is much ing over the noise must restore the original master equa- less tractable than the Gaussian noise typically associ- tion; and (iii) it must yield the correct autocorrelation ated with steady photocurrents. For these reasons, we function for the photocurrent. As in Sec. III, the first wish to develop a quantum theory of intensity measure- two conditions suggest a stochastic master equation of ments appropriate for systems with large, stable inten- the form 5188 H. M. WISEMAN 47

P (t) = [& + v 7((t)2 ]p (~) (6.3) will generate valid density-matrix evolution. This will be shown to be the case for an ideal laser. The proof relies where 2, is the generator of the original master equation, upon keeping only terms of leading order in the mean and 5 is the nonlinear superoperator for intensity mea- photon number. surements which is to be determined. The photocurrent autocorrelation function which we require is [19] VII. MONITORING THE REGULAR LASER

E(I.(t + ~), I(t)) We first investigate the effect of direct photodetection on the regularly pumped laser state, using the theory of = q Tr(algae [ap()!)af —(aia)(g) p(g)) j the previous section. Converting Eq. (6.7) into a Fokker- +g(aia), (t)6(~). (6.4) Planck equation for the Wigner function and adding the ideal laser terms gives

Within the approximations mentioned above, it is per- O 1 O 1 1 O W, (t) = ( — )+- ( + +- missible to replace the final shot noise term by rIpb(w). ~On )~, the method of Sec. we can show from Using III, Eq. (6.2) 0 1 c)' — n — ') — that the stochastic master equation Eq. (6. gives the ~ n, (), n 3) ((t) + n+ Bnz result p ( Bn 2 x W, (t), (7.1) E(I,(t + ~), I(t)) = il ~p Tr(a ae [2p(t)]) + rjpb(7. ). where n, (t) = (aia), (t). Now we linearize the (6.5) measurement-induced terms as above, and discard terms We see immediately that the required superoperator is of order 1j~p, . In doing this we make the crucial assump- defined by tion that the photon-number variance of the conditioned 1 state is of the same order as the mean. Since the phase Zp = apai —Tr(cpa') p (6 6) dependence of the Wigner function is not influenced by the intensity measurement, we remove this degree of free- That is to say, with the photocurrent given by Eq. (6.2), dom to obtain stochastic evolution equation for the conditioned sys- the O 1 O tem state is W, (n, t) = (n —p) + —p(2+ q) (),(t) = (L+ ((t) (Z(a) —(a a), (t))) P, (t). (6.7) — o) ') ~ n ((t) n, (t) + p, [ W, (n, t). On) Having "derived" this master equation, we must em- (7.2) phasize that it is not valid in general. Although it obviously preserves Hermiticity and trace, it does not in Using the method of Appendix A, we find that this general preserve the positivity of density operators. For equation has a Gaussian solution. The mean n and vari- example, consider an initially pure state with mean pho- ance cr~(t) of this distribution obey the following equa- ton number p, in a simply decaying cavity. It is easy to tions: show that, to leading order in the infinitesimal time dt, n, = —n, + p, + (~) — (7.3) 2dlV (t) (t) . .(t) ((~) T [~(«)'] = 1+ ((~') (a) —(~'a)) (6.8) ' (t) = -2 '(~) + ~(2+ ~) —— '(t) —n. (t) . p . where all of the quantum averages are evaluated at t = 0 (7.4) and we have assumed g = 1. Unless the initial state is a coherent state, the infinitesimally evolved state has a 50%%uo These equations are similar to those for the moments of chance of being nonpositive, with Tr[p(«) ] & 1. This the phase distribution in Sec. IV, in that the noise only shortcoming is a consequence of the fact that this theory appears in the equation for the mean. Note the difFerent is intended to apply to coarse-grained measurements. A effect on the conditioned mean depending on whether the stochastic master equation describing fine-grained mea- state is super- or sub-Poissonian. For a super-Poissonian surements [such as (3.3) with il = 1] is equivalent to state (o' & n), the conditioned mean photon number in- a stochastic Schrodinger equation for a state vector [6], creases when a higher than average intensity is measured providing that the nondamping evolution is unitary. The at the photodetector [((t) & 0]. This is what is expected density matrix formed from a state vector is of course from a classical measurement; the higher photocurrent positive definite, so fine-grained measurements will al- indicates that; the true mean photon number of the sys- ways give valid master equations. Despite this problem, tem is larger than we had previously thought, and so our Eq. (6.7) is useful for those applications for which it estimate (n, ) is adjusted upwards. For a sub-Poissonian was intended. That is, if the system has a well-defined state, this intuition fails in that a higher photocurrent stationary mean photon number (with variance of the causes the conditioned mean to decrease. This is pre- same order as the mean), the stochastic master equation cisely the mechanism by which squeezed light gives rise STOCHASTIC QUANTUM DYNAMICS OF A CONTINUOUSLY. . . 5189 to a subshot noise photocurrent, and highlights the dif- interpretation in terms of two components: the quan- ference between classical and nonclassical states. The tum variance o.2 around each conditioned mean n„and explanation lies in the fact that an above average pho- the classical (ensemble) variance due to the random walk tocurrent is caused by an increase in the number of pho- of nc under photodetection. The relative proportion of tons leaving the cavity. If the photon-number variance is these two contributions depends on the efFiciency of the sufficiently small (less than the mean), then the eKect of detection. this in dropping the mean photon number in the cavity We now examine heterodyne detection on the regu- overrides the "classical" efFect described above. lar laser. Converting the heterodyne stochastic master The deterministic equation for the variance will cause equation (3.3) into a linearized stochastic Fokker-Planck the photon-number variance to collapse rapidly to its equation for the signer function gives steady-state value of cr 1 g2 1 1 g2 ~.(t) = (n —p) + —p(2+ q); +- ~' = —[—1+re+ QI+qrl]. (7.5) Bn 2 'rt + "(t) n —n, (t) + p, If ri = 1 (perfect detection), we have simply (, cr' = p, gl+q, (7.6) +/2prl g~(t) p —p, (t) + W, (t), 4p &V which is to be compared with the nonconditioned vari- (7.11) ance of p(1+q/2). From this it is obvious that the conditioned variance is less than or equal to the unconditioned = where P,(t) is the mean phase of the Wigner function, variance for g 1. In fact, this is true for any value of and we now have two real white-noise terms, and as ~ 0 we recover the nonconditioned value. g, = g If q 0, then the conditioned variance is equal to the (,"(t) = Re[v 2e'~ ('l((t)] (7.12) nonconditioned variance p, . This is as we concluded ear- 2e'~ ( lier: for a Poissonian pumped laser, intensity measure- (~(t) = 1m[v l((t)], (7.13) ments have no effect. For a sub-Poissonian pumped laser where is the complex white noise in Eq. (3.3). It (q 0), the nonclassical nature of the intracavity field is ((t) ( is evident that, in this linearized regime, the efFect of enhanced in the conditioned state. In particular, if ~ 1 q heterodyne detection on the photon-number distribution and —+ —1, then we get o —+ 0. In this regime, our q is same as effect direct with approximations break down, as we have assumed that the the the of photodetection, the efficiency reduced a factor of 2. This is not un- variance is always of the same order as the mean. Never- by as the heterodyne contains theless, it indicates that under ideal conditions, the ratio expected, photocurrent equal information about the intensity and phase of the cavity of the variance to the mean in the conditioned state will be arbitrarily small. mode. The above results for direct photodetection thus over heterodyne detection with the replacement of Substituting the steady-state value for the conditioned go to variance into the equation (7.3) for the conditioned mean g by rl/2. The effect of heterodyne detection on the phase is the gives same as it was in the Poissonian laser. To see this we = — — derive the coupled stochastic differential equations, again n, n, (t) + p+ 1+ pl+ qg ((t). (7.7) using the method of Appendix A

If = 0, then there is no noise in the mean, as expected. g (7.14) The same applies if q = 0. In general, n, undergoes a random walk with a linear restoring force. That is, it obeys an Ornstein-Uhlenbeek process [18]. Averaging over the ensemble of conditioned states, we have the fol- (7.15) lowing steady-state expectation values where V, is the conditioned Wigner phase variance. E(n, ) = p, (7.8) (t) These are identical to the equations for the Poisson laser when one remembers that = As- E(n2 —ps) = —[1 —Ql + qg+ 2rqri]. (7.9) (4.7) V, (t) U, (t) +1/4p. suming that g = 1, we have a conditional phase variance of The total uncertainty in the photon number in the ensemble of conditioned states is equal to the photon-number variance in each conditioned state (7.5) plus the ensemble (7.16) variance in the mean photon number (7.9), Meanwhile, putting rt = 1/2 into Eq. (7.5) gives the — = — o + E(ri~ tJ2) y. (1+ ) . (7.10) heterodyne detection photon-number variance under the same conditions as This is precisely equal to the unconditioned variance in the photon number (5.9). Here, this variance has a new cr2 = p(v2 —1) = 0.41p, (7.17) 5190 H. M. WISEMAN 47 where we have put q = —1. Thus the conditioned state 8[at], can be added to the list of classical processes such for a perfectly regular laser under heterodyne detection as damping, detuning, and coherent driving. is highly nonclassical, and has a Wigner phase-space area Although the results we have derived for the condi- of tioned state of a laser are helpful pedagogically, it is rea- sonable to ask what practical use they have. Usually, the 2 — conditioned state of a system is of little interest to experi- QV~o. = 2 3(~2 1) 0.56. (7.18) menters and some would even doubt whether the concept This is only just above the minimum of 0.5 allowed by the is meaningful. In fact, the conditioned states such as we Heisenberg uncertainty relations for number and phase. have investigated are meaningful experimentally, because they are precisely those states which can be stabilized by using the measured photocurrent (on which they are VIII. DISCUSSION conditioned) in a feedback loop. The proof of this statement will appear in future work. This implies that the The stochastic dynamics of a continuously monitored steady-state conditioned variances which we derived here laser are essentially given in Sec. VII. There we consid- represent the best achievable steady-state variances un- ered a laser with arbitrary pump statistics under both di- der feedback. In summary, the practical significance of rect photodetection and heterodyne detection. The pic- this work is that conditioning is realized by feedback. ture which emerged is this: With no measurement, the unconditioned state of the laser has a well-defined intensity (with variance determined by the regularity of the ACKNOWLEDGMENT pump), but completely undefined phase. Under heterodyne detection, the laser state is collapsed rapidly (on I wish to thank G.J. Milburn for helpful advice and the time scale of the cavity linewidth) to a state with a encouragement. well-defined phase. For an ideal laser with unit efficiency detection, the steady-state phase variance is three times that of a coherent state. The conditioned mean phase undergoes a random walk. The source of randomness is APPENDIX A: GAUSSIAN SOLUTION TO the shot noise in the heterodyne photocurrent on which STOCHASTIC FOKKER-PLANCK EQUATION the state of the system is conditioned. For a Poissonian pumped laser, heterodyne detection Consider a nonlinear Fokker-Planck equation in one has no effect on the conditioned photon-number variance, dimension with an Ito stochastic term of the form but in all other cases the conditioned variance is less [18] than the unconditioned variance. In the con- particular, P(x, t) = ~A((t) [x —x(t)]P(x, t), (A1) ditional number squeezing in a regularly pumped laser is enhanced. The conditioned mean photon number also where ((t) represents real Gaussian white noise and has a random trajectory related to the noise in the photocurrent. Unlike heterodyne detection, direct photode- x(t) = Cx P(x, t)x (A2) tection has no effect on the phase of the laser. To com- pensate, its effect on the amplitude variance is enhanced We call Eq. (Al) a Fokker-Planck equation because, if by a factor of 2 over that of heterodyne detection. Unit we assume a solution of the form efficiency direct detection of a perfectly regularly pumped 1 laser causes the conditioned photon-number variance to P(x, t) = exp( —[x —x(t)] /2U(t)), (A3) become arbitrarily smaller than that of a coherent state. +2irU(t) An aspect of interest in the analysis of the condition- then we can rewrite as ing of the laser state on external photodetection is that it Eq. (Al) reinforces the distinction between classical and nonclas- — sical states. By "classical" states we mean those with a P(x, t) = v A((t) U(t) P(x, t), (A4) positive Glauber-Sudarshan P function [2,3]. The state of an open quantum system conditioned on external pho- which has the form of a nonlinear drift term. However, todetection (whether it be direct detection, or with the it will be seen to have an effect on both the variance and addition of a local oscillator) will not be nonclassical un- the mean of the distribution. To see this we note that less the nonconditioned state is nonclassical. This can be the rules of Ito stochastic calculus give viewed either as a statement about the nature of external v AdW(t) — photodetection (photodetection cannot produce nonclas- 1+ [x x(t)] sical states) or about the definition of coherent states (coherent states are those states which are uninfluenced =exp AdW t 2; —2 t ——dtx —x t by photodetection). In this context, it is worth noting 2 that the ideal model of a laser which we developed contains the definition of another quantum optical process (A5) which is inherently classical. This process of incoherent excitation, which we have denoted by the superoperator Applying this to the ansatz (AS) and rearranging gives STOCHASTIC QUANTUM DYNAMICS OF A CONTINUOUSLY. . . S191

1 1 A P(x, t+dt) = —I* —*]~)I' + — W~dw]* —*-]~]]) +2~U(t) 2U(t) «+ 1 exp([z —x(t + dt)] /2U(t + dt) +27rU(t + dt) ), (A7)

I where we have defined That is to say, we have the following approximate superoperator correspondence: x(t + dt) = x(t) + v AU(t)dW, (A8) = —AU(t)'dt. 1 1 B2 1 U(t + dt) U(t) (A9) &[a'1 'p W. —+ BaBn* (B7) E, ]n/ + 2/n[ 2[n. ] That is to say, the nonlinear stochastic equation (Al) ], does have a Gaussian solution. The mean obeys the Next, it is easy to show that stochastic equation expected from Eq. (A4), and the variance obeys a deterministic equation which causes it al- —1 ——1 B 1 B &[a'1p I ln '+ ways to shrink. 2 2Bn 2Bn* (B8) APPENDIX B: WIGNER FUNCTION FOKKER-PLANCK EQUATION FOR Thus, to second order in the mean photon number we REGULARLY PUMPED LASER have

/' We wish to convert the regularly pumped laser master 1 B —1 1 I I+cc equation .+, . 0 1 2 (B9) (B1) BnBn* [n[2 into a Fokker-Planck equation for the Wigner function. and Recall that we are using the notation 1 8 1 0 1 — c.c. c. — C]a~] p + + ~ ~ +c. t [at] = +[at]A[at] 1, &[a] = J'[a] —A[a], (B2) 4 Ba; n 2n* Bn2 n* ) 0 2 where + (B10) BaBn* [n~2 A[c]p = 2(ctcp+ pctc), Q[c)p = cpct. (B3) Finally, it is a standard result that First, consider the density operator o. defined by

= —,'(aa'o+ o.aat). 1 B B ~ B p (B4) 'D[a]p —+ — n+. n*+ W. 2 Bn Bn* BnBn* (B11) If W(n, n*) is the Wigner function for p and V(n, n') is that of o, then standard operator correspondences [19] Putting all of these results together gives the Fokker- give Planek equation quoted in the text (5.12)

W= a, + —— V. (B5) 2 —,4 BaBn* lV + c.c. Bn 2n 2 2 4[n]2n* Assuming that the mean photon number is very large, we can invert this differential operator to second order in QP 2 +- + [n~ to find 2 Bn 4n' Bn' 4a.

1 1 B2 1 s + (B6) ~+(i+-,') ", (B12) 2[n[ BnBn' 2[n/ +, , )w.

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