19. Position Measurement¶
PDF pages 865–906
19.1.1 Positive Maps¶
Chapter 19 Position Measurement Here we will study the continuous observation of a Hermitian observable, namely the position of a quantum particle. We will do so fairly abstractly, but then give a physical example of how a position measurement can arise in atomic resonance fluorescence. 19.1 Prelude: General Form for the Master Equation Before working out another continuous measurement process, we can ask the question, what is the most general form of the measurement master equation when the measurements involve Gaussian noise? A simple but nonrigorous argument1 that establishes the general form for the unconditioned master equation, and then extend it to examine stochastic master equations (SMEs). Thus, we will see that the form of the (Markovian) SME involving Wiener noise is quite constrained, and it is intuitively easy to adapt the SME to many different measurement processes simply by choosing the correct measurement operator (which we denote below by c). 19.1.1 Positive Maps Under unitary (unconditioned) evolution, the Schrödinger equation tells us that in a short time interval dt, the state vector undergoes the transformation
1 −iH ¯h dt
(19.1) where H is the Hamiltonian. The same transformation applied to the density operator gives the Schrödinger– von Neumann equation (from Section 4.1):
1 −iH ¯h dt \rho 1 + iH ¯h dt
¯h[H, \rho] dt. (19.2) To be physical, any transformation of the density operator must be completely positive. That is, the trans- formation must preserve the fact that the density operator has only nonnegative eigenvalues. This property guarantees that the density operator can generate only sensible (nonnegative) probabilities. (To be more precise, complete positivity means that the transformation for a system’s density operator must preserve the positivity of the density operator—the fact that the density operator has no negative eigenvalues—of any larger system containing the system.) It turns out that the most general form of a linear, completely positive 1S. L. Adler, ‘‘Derivation of the Lindblad generator structure by use of the Itô stochastic calculus,’’ Physics Letters A 265,
19.1.2 Lindblad Form¶
Chapter 19. Position Measurement transformation is2 \rho −\rightarrow X n
n, (19.3) where the An are arbitrary operators. The Hamiltonian evolution above corresponds to a single infinitesimal transformation operator A = 1 −iH dt/¯h. 19.1.2 Lindblad Form Now let’s examine the transformation for a more general, stochastic operator of the form A = 1 −iH ¯h dt + b dt + c dW, (19.4) where b and c are operators. We will use this operator to ‘‘derive’’ a Markovian master equation, then indicate how it can be made more general. We may assume here that b is Hermitian, since we can absorb any antihermitian part into the Hamiltonian. Putting this into the transformation (19.3), we find
dW, (19.5) recalling that [A, B]+ := AB + BA is the anticommutator. We can then take an average over all possible Wiener processes, which again we denote by the double angle brackets \langle \langle \rangle \rangle . To compute the ensemble average, we again use the property \langle \langle \rho dW\rangle \rangle = 0 of It¯o calculus, so that
(19.6) Since the operator \langle \langle \rho\rangle \rangle is an average over valid density operators, it is also a valid density operator and must therefore satisfy Tr[\langle \langle \rho\rangle \rangle ] = 1. Hence we must have dTr[\langle \langle \rho\rangle \rangle ] = Tr[d\langle \langle \rho\rangle \rangle ] = 0. Using the cyclic property of the trace, this gives the constraint Tr
= 0. (19.7) This holds for an arbitrary density operator only if
2 . (19.8) Thus we obtain the Lindblad form3 of the unconditioned master equation (averaged over all possible noise realizations):
(19.9) As before, we have defined the Lindblad superoperator
, (19.10) where ‘‘superoperator’’ refers to the fact that D[c] operates on \rho from both sides. It is worth reiterating here that the c\rhoc\dagger term results from the dW part of the transformation, and thus this term cannot be represented by a Hamiltonian transformation, even if the Hamiltonian is non-Hermitian, as we noted in Section 5.5.3. This is the most general (Markovian) form of the unconditioned master equation for a single dissipation process. Different choices for the operator c thus give the quantum backaction (disturbance) for different measurement processes. 2K.-E. Hellwig and K. Kraus, ‘‘Operations and Measurements. II,’’ Communications in Mathematical Physics 16, 142 (1970); Benjamin Schumacher, ‘‘Sending entanglement through noisy quantum channels,’’ Physical Review A 54, 2614 (1996) (doi: 10.1103/PhysRevA.54.2614). 3G. Lindblad, ‘‘On the generators of quantum dynamical semigroups,’’ Communications in Mathematical Physics 48, 199 (1976) (doi: 10.1007/BF01608499).
19.1.4 Generalization¶
19.1 Prelude: General Form for the Master Equation 19.1.3 Stochastic Terms The full transformation from Eq. (19.5) then becomes
dW. (19.11) This is the linear SME, which we will discuss again elsewhere. We know from our treatment in Section 18.2 of homodyne detection (where c −\rightarrow \sigma) that this equation is not ‘‘complete,’’ since it is missing the nonlinear term. The problem is that this form of the master equation does not in general preserve the trace of the density operator, since the condition Tr[d\rho] = 0 implies Tr \rho
dW = 0. (19.12) We could interpret this relation as a constraint4, on c, but we will instead keep c an arbitrary operator and explicitly renormalize \rho + d\rho by adding a term proportional to the left-hand side of (19.12). The result is the nonlinear form
(19.13) where again the measurement superoperator is
\rho. (19.14) This corresponds to using the normalizing transformation \rho −\rightarrow
(19.15) instead of the unnormalized transformation that we considered,
(19.16) to first order in dt. The normalized transformation has exactly the form of a POVM-type reduction as we consider below. When c = \sqrt \Gamma\sigma, we recover precisely the master equation for homodyne detection of spontaneous emission. In general, c can be chosen differently to model different continuous measurement processes. 19.1.4 Generalization More generally, we may have any number of measurements, or output channels, happening simultaneously. The result is
X n (D[cn]\rho dt + H[cn]\rho dWn) . (19.17) This is the same as Eq. (19.13), but this time summed (integrated) over multiple possible measurement op- erators cn, each with a separate Wiener noise process independent of all the others. This simply corresponds to having multiple terms in the general positive map (19.3). In view of the arguments from our treatment of detector efficiency in homodyne detection in Sec- tion (18.2.8), when the measurements are inefficient, we have5
X n
(19.18) 4S. L. Adler, op. cit. 5This form is close to the most general form of the master equation, but can still be generalized further. See H. M. Wiseman and L. Diosi, ‘‘Complete parameterization, and invariance, of diffusive quantum trajectories for Markovian open systems,’’ Chemical Physics 268, 91 (2001) (doi: 10.1016/S0301-0104(01)00296-8).
19.2.1 Discrete, Finite Spaces¶
Chapter 19. Position Measurement where \etan is the efficiency of the nth detection channel. The corresponding measurement record for the nth process can be written (with an arbitrary normalization) as
n
dt + dWn
. (19.19) Again, for homodyne detection, we recover the right results if we let c = \sigma and interpret drn/2\Gamma as a rescaled measurement record. 19.2 A Second Prelude: Positive-Operator-Valued Measures To help handle generalized measurements, we will now introduce the somewhat mathematical concept of a positive-operator-valued measure (POVM). By referring to generalized measurements, we mean to differ- entiate these measurements from the usual projective, or von Neumann, measurements, which is what you normally find in introductory quantum-mechanics texts. The usual description goes like this: for a quantum system in state |\psi\rangle , a measurement of the observable Q leaves the system in an eigenstate |q\rangle of Q with probability \langle \psi|Q|\psi\rangle , in which case the ‘‘result’’ of the measurement is the eigenvalue q. We can see that this notion of a measurement is lacking in two situations. First, it does not properly describe the situation in photodetection of atomic radiation, where each detection event results in the loss of energy (i.e., the atom is always found to be in the ground state), and we gain information even during instants when a photon is not detected. Thus, POVMs are crucial to the formal definition of a continuous measurement. The second situation is when the observable is the position operator, where eigenstate collapse is unphysical: a position eigenstate is a state of infinite energy. POVMs allow us to define an imprecise or partial measurement of an observable, which will be a stepping stone on the way to defining a continuous measurement of position. 19.2.1 Discrete, Finite Spaces Consider a discrete, finite Hilbert space of dimension N. That is, the Hilbert space is spanned by the set of eigenstates
(19.20) of the observable Q. Then we can define a positive-operator-valued measure (POVM) as a set of positive-semidefinite operators Ω\dagger qΩq that sum to the identity operator: Nq X q=1 Ω\dagger qΩq = 1. (19.21) Note that we are writing the qth positive operator Ω\dagger qΩq as a factorization in terms of the Kraus operator Ωq, since any positive operator always has such a factorization. We also note that the number Nq of positive operators is not necessarily the same as the dimension N of the Hilbert space. Now the important physical point here is that a POVM defines a quantum measurement on the Hilbert space. The qth possible outcome of the measurement is that the state vector changes according to the replacement
rD Ω\dagger qΩq E, (19.22) or in terms of the density operator, \rho −\rightarrow
q
q]
q D Ω\dagger qΩq E. (19.23)
19.2.2 Measure¶
19.2 A Second Prelude: Positive-Operator-Valued Measures That is, in the qth outcome, the state is ‘‘hit’’ by the operator Ωq and then renormalized if necessary. The probability that the qth outcome occurs is
q] =
Ω\dagger qΩq
. (19.24) The (classical) ‘‘result’’ of the quantum measurement in this case is simply q (or some physically meaningful function of q). This notion may seem rather abstract, but we can note that the usual projective measurement comes out as a special case of the POVM-based measurement. In particular, the usual measurement arises from a projection-valued measure, where we partition the Hilbert space according to a set of (Hermitian) projection operators
(19.25) that also sum to the identity: N X q=1 P 2 q = 1. (19.26) Of course, P 2
by taking Ωq = Pq and Nq = N. Then the standard projective measurement of the observable Q results in the qth outcome of a reduction to the qth eigenstate |q\rangle ,
q P 2 q
(19.27) or in terms of the density operator, \rho −\rightarrow
q
q ]
q
P 2 q
(19.28) This outcome happens with probability
q ] =
P 2 q
(19.29) which for a pure state |\psi\rangle becomes the familiar Born rule
(19.30) Thus, the POVM-based measurement above is a reasonably straightforward generalization of the usual projective measurements, at least when the standard measurements are cast in the proper way. 19.2.2 Measure Why is a POVM called a ‘‘POVM’’? The answer requires an excursion into mathematics, and so the short answer, if you feel the need to skip forward, is that a measure is usually something that assigns numbers to sets, and so a positive-operator-valued measure is a measure that instead associates positive operators with sets, and thence probabilities to the same sets via the expectation value as above. To really answer this question, we need to define what we usually mean by a measure, and then adapt it to the operator case. Informally, a measure is a rule for assigning numbers to subsets of some set, or space. This is a very useful notion in probability theory, where you would consider the set of all possible outcomes or events, and the measure would assign probabilities to each outcome or collection of outcomes. Alternately, a measure is an abstraction of the notion of volume, where the measure represents the ‘‘volume’’ of subsets of the main set. Before formally defining a measure, though, we should first note that for a given space, it is problematic to try to define a measure on every subset. Instead, we will define the measure on only a limited collection of subsets, chosen to make the definition of the measure consistent. Formally, this collection is a \sigma-algebra, which we define as a collection S of subsets of the space X such that:
Chapter 19. Position Measurement 1. The empty set is included: ∅\in S . 2. Countable, disjoint unions are included (with countable here meaning finite or countably infinite): if U ⊂S with A ∩B = ∅for any A, B \in U , and U is countable, then [ A\in U A \in S . (19.31) 3. Complements are included: if A \in S , then X −A \in S . Any element of a \sigma-algebra is said to be a measurable set. This definition can be contrasted with the possibly familiar definition for a topology on a space X, which is a collection T of subsets of X such that:6 1. The empty set and the whole space are included: ∅\in T , X \in T . 2. Arbitrary unions are included: if U ⊂S , then [ A\in U A \in S . (19.32) 3. Finite intersections are included: if U ⊂S with U finite, then \ A\in U A \in S . (19.33) Any element of a topology is said to be an open set, while the complement of an open set is said to be a closed set. Thus, while topologies contain in general only open sets, \sigma-algebras contain both open and closed sets. For example, on the real line R, the standard topology is the topology consisting of all open intervals of the form (a, b) and all possible unions of such intervals (and the empty set). It turns out there is a unique \sigma-algebra associated with the standard topology, which is the smallest \sigma-algebra containing it. This is called the Borel \sigma-algebra on R, which would contain all open intervals as well as all closed intervals of the form [a, b] (and many other sets). The notion of a \sigma-algebra may not be intuitively clear at this stage, but the definition is basically concocted to make the definition of measure work out, as we will now see. A measure is a function µ : S −\rightarrow [0, \infty] defined on a \sigma-algebra S on a space X, which satisfies
- The measure for countable, disjoint unions adds: if U ⊂S with A ∩B = ∅for any A, B \in U , and U is countable, then µ [ A\in U A ! = X A\in U µ(A). (19.34) These two requirements are sensible considering our analogies to probabilities and volumes, and we can also see how the requirements for a \sigma-algebra guarantee that we don’t have any problems in defining a measure (the last axiom for a \sigma-algebra imposes the sensible constraint that if A is a measureable subset, then so is X −A). Note that the point \inftyis explicitly included in the range of a measure, which is intuitively a ‘‘good’’ measure for something like the entire real line. Also, strictly speaking, we have defined a positive measure, since we have only allowed nonnegative values in the range of µ. As an example of measure, the Lebesgue measure on the real line is defined on the Borel \sigma-algebra. We can define it in several cases as follows: 1. It turns out that any open set A can be written as the union of a countable set of open intervals (aj, bj), in which case the Lebesgue measure of A is the sum of the interval lengths:
X j (bj −aj). (19.35) 6For further reading, see, e.g., James R. Munkres, Topology: a First Course (Prentice-Hall, 1975).
19.2.3 General Definition¶
19.2 A Second Prelude: Positive-Operator-Valued Measures 2. It turns out that any closed set B can be written as a closed interval [a, b] with the union of a countable set of open intervals (aj, bj) removed from it, B = [a, b] − [ j (aj, bj), (19.36) where every aj > a and every bj < b, in which case the Lebesgue measure of B is the length of the closed interval minus the Lebesgue measure of the removed component:
X j (bj −aj). (19.37) 3. For any other set C in the Borel \sigma-algebra, the Lebesgue measure is the infimum (greatest lower bound) of the set of Lebesgue measures of all open sets containing C:
(19.38) Note that there exist sets that do not have Lebesgue measures according to the above definitions, and thus they are excluded by considering only the \sigma-algebra. The Lebesgue measure is useful in that it extends the notion of length to more complicated and subtle sets: the set of rational numbers, being countable, is a set of Lebesgue measure zero on the real line; and the Cantor middle-thirds set, a fractal set constructed by starting with the interval [0, 1], removing the open ‘‘middle third’’ interval (1/3, 2/3), removing the middle-thirds of the two remaining closed intervals, and so on ad infinitum, is an uncountable set but of zero Lebesgue measure. For measurements, the concept of a probability measure is more useful, and it is simply that of a measure, but where the range of the measure is [0, 1] rather than [0, \infty], with a measure of the whole space being unity. For example, the Lebesgue measure on the space [0, 1] is a probability measure, and corresponds to a uniform probability density on the same interval. 19.2.3 General Definition Now with the above mathematical concepts, we can now give a more general definition of a POVM than in the finite case above. In more general terms, a positive-operator-valued measure (POVM) defined on a \sigma-algebra S on a space X is a function \Pi that takes as values positive semidefinite, Hermitian operators on a Hilbert space H such that for any |\psi\rangle \in H , the function µ : X −\rightarrow [0, 1], defined by
(19.39) for any measurable subset A of X, defines a probability measure on S . In particular, this implies that \Pi(X) is the identity operator, which is the generalization of the sum rule (19.21). Thus, the POVM associates positive operators with measurable subsets of the space of outcomes, which are then associated with probabilities by appropriate expectation values. In this way, we can define a family of probability measures, ‘‘parameterized’’ by the quantum state. We could, of course, write the probability measure more generally in terms of the density operator as
(19.40) Incidentally, a trace of this form is (for a Hilbert space of dimension larger than two) the only way to construct a quantum probability measure; this is essentially the content of Gleason’s theorem.7 7Andrew M. Gleason, ‘‘Measures on the Closed Subspaces of a Hilbert Space,’’ Journal of Mathematics and Mechanics 6, 885 (1957) (doi: 10.1512/iumj.1957.6.56060).
19.2.4 Realization¶
Chapter 19. Position Measurement 19.2.4 Realization It is important to note that measurements induced by POVMs, while generalizing projective measurements, don’t introduce anything fundamentally new to quantum mechanics: any of these more general measurements can be realized by introducing an auxiliary system (ancilla), performing a unitary transformation on the combined system, and then perform a projective measurement on the ancilla. Thus, generalized measurements correspond to indirect measurements, where information about a system comes from projective measurements on the ‘‘environment’’ with which the system has interacted (and thus become entangled with). This result is known as Naimark’s theorem (or Neumark’s theorem),8 and we will only sketch the argument for the finite case here. Starting with the system in the state |\psi\rangle , we will extend the Hilbert space to contain the environment, whose dimension is equal to the number of Kraus operators defining the POVM, |\psi\rangle −\rightarrow |\psi\rangle |0E\rangle ≡|\psi 0E\rangle . We will assume the environment to always start in a particular state that we label |0E\rangle . Note that we are assuming a pure state for the system, which we may as well do as long as we are extending the Hilbert space by invoking purification (Section 4.4.5). We can thus define an operator U that acts on the composite state as
X q
X q q
q
|qE\rangle , (19.41) where the Kraus operators Ωq only operate on the original system, and the |qE\rangle environment states are orthogonal. In the last step we have written the part of the state of the original system as a normalized state, leading to explicit coefficients of the superposition. We can also see explicitly how the system and environment are entangled after the operation U. Now computing the norm of the transformed composite state,
X qq′
E\rangle = X q
(19.42) so that U preserves the norm of states in the subspace of the original system. The operator U is thus unitary on this subspace, but is not fixed uniquely by the above argument. In principle, the action of U on the environment can be chosen to make U unitary on the composite Hilbert space. Basically, this is because taken as a matrix, the columns of U span the subspace of the original system (i.e., a subset of them form an orthonormal basis), and the extra degrees of freedom (elements of the extra rows) in expanding U to the composite Hilbert space may then be chosen to make the columns of U form an orthonormal basis on the entire composite space. Now after the transformation, a projective measurement of the state of the environment leads to the result |qE\rangle with probability Tr
(19.43) Furthermore, the projection of the environment into state |qE\rangle induces the transformation
q
(19.44) on the original system. Thus we have constructed the POVM-based measurement based on the larger projective measurement. 8Asher Peres, Quantum Theory: Concepts and Methods (Springer, 1995), Section 9-6, p. 285. For a similar argument to what we present here for the unitary representation of linear positive maps, see Benjamin Schumacher, op. cit.
19.2.6 Example: Gaussian Projectors¶
19.2 A Second Prelude: Positive-Operator-Valued Measures 19.2.5 Example: Spontaneous Emission As an example of a POVM, we return to the stochastic master equation for photodetection of atomic resonance fluorescence with quantum jumps from Section 18.1:
¯h[H, \rho]dt −\Gamma
dN. (19.45) In any given time interval of duration dt, there are only two possible outcomes: no photon is detected, or one photon is detected. We can define this evolution in terms of a POVM as follows. Let U(dt) denote the evolution operator for the combined atom–field system. Before each infinitesimal time interval, the field starts in the vacuum state |0\rangle , and after each infinitesimal time interval, the detector projectively measures the field and registers a detection event if a photon is emitted into any mode. Since the detector does not distinguish modes, we will simply denote the field state as |1\rangle in the case of an emitted photon. Then the two ‘‘jump operators’’ for the two measurement outcomes are9
¯h dt −\Gamma
\sqrt \Gamma dt \sigma. (19.46) In the case of no photon detected, the state is transformed according to \rho −\rightarrow
0(dt) Tr h
0(dt)
¯h[H, \rho] dt −\Gamma
\rho dt, (19.47) keeping terms to first order in dt, and in the case of a detector click the state is transformed according to \rho −\rightarrow
1(dt) Tr h
1(dt)
(19.48) These two transformations correspond exactly to the transformations induced by the SME (19.45) in the cases dN = 0 and dN = 1, respectively. The probabilities also work out as expected. For example, a photon is detected with probability
h
1(dt) i = \Gamma
dt, (19.49) and the probability for not detecting a photon is the complement of this, as seen by taking the appropriate trace using Ω0(dt). Notice that this POVM tends to drive the atom towards the ground state, as compared to the uncon- ditioned Hamiltonian evolution (and for either possible outcome Ω0,1). By involving the atomic annihilation operator, we see in this case that the POVM generalizes projective measurements by modeling dissipation due to the measurement process. In the case at hand, the physical origin of the dissipation in the case at hand is absorption of radiated photons by the photodetector. 19.2.6 Example: Gaussian Projectors POVMs can also generalize projective measurements to model partial or imprecise measurements. Partial measurements leave some uncertainty in the measured observable, whereas projective measurements leave the system in a state where the observable is perfectly defined—that is, an eigenstate of the observable. As a simple example, we can model partial measurements by defining the measurement operators Ωq to be Gaussian-weighted sums over projection operators for the discrete set of eigenstates |q\rangle (q \in Z) of the observable Q: Ωq = 1 N \infty X j=−\infty
(19.50) 9H. M. Wiseman, ‘‘Quantum Trajectories and Quantum Measurement theory,’’ Quantum and Semiclassical Optics 8, 205
Chapter 19. Position Measurement Here, N 2 := \infty X j=−\infty
(19.51) so that \infty X q=−\infty Ω\dagger qΩq = 1, (19.52) as required for the operators to form a POVM. The Gaussian weights lead to having only partial infor- mation about Q after the measurement. For example, in a highly uncertain mixed state, where \langle q|\rho|q\rangle is approximately the same for any q and \langle q|\rho|q′\rangle = 0 for any q̸ = q′, the measurement leads to the collapse \rho −\rightarrow
q
q] \approx 1 N X j
(19.53) The qth possible final state is thus peaked about the eigenvalue q, and additionally has an uncertainty
In the limit \kappa −\rightarrow \infty, the measurements here reduce to the usual projective measurements. Thus, for large \kappa, the variance in the measurement results (taken over an ensemble of measurements on identically prepared systems) is dominated by the uncertainty in the quantum state, while for small \kappa, the measurement variance is dominated by the uncertainty introduced by the measurement operators Ωq. This distinction divides two categories of measurments, strong measurements where \kappa is large, and weak measurements, where \kappa is small.10 We can also generalize these Gaussian projectors to the continuous-variable case. For example, for a position measurement, the properly normalized measurement operators have the form
\kappa 2\pi 1/4 Z \infty −\infty
(19.54) Again, if this operator is applied to a an initially uncertain state (such as a momentum eigenstate), the resulting position variance in the collapsed state is 1/\kappa (i.e., the uncertainty is 1/\sqrt\kappa). In what follows, we will consider sequences of weak position measurements of this form, and thus construct continuous quantum measurements of position. For this it is useful to consider the product of two operators,
(2\pi)2 1/4 Z \infty −\infty dx′ Z \infty −\infty
=
(2\pi)2 1/4 Z \infty −\infty
=
(2\pi)2 1/4 exp
Z \infty −\infty dx exp "
2#
(19.55) which corresponds to a sequence of two Gaussian position measurements, the first of strength \kappa and the second of strength \kappa′, with measurement outcomes \alpha and then \alpha′, respectively. This operator product is still Gaussian, but it is not normalized properly in the sense that Ω(\alpha) is normalized (note that the norm vanishes if \alpha −\alpha′ becomes large), but we can see from its form that applying this operator to an initially uncertain state gives 1/(\kappa + \kappa′) for the resulting position variance of the state. Hence, a sequence of two Gaussian measurements is effectively equivalent to a single Gaussian measurement, where the strength is 10Yakir Aharonov, David Z. Albert, and Lev Vaidman, ‘‘How the result of a measurement of a component of the spin of a spin- 1 2 particle can turn out to be 100,’’ Physical Review Letters 60, 1351 (1988) (doi: 10.1103/PhysRevLett.60.1351). See also comments by A. J. Leggett, ‘‘Comment on ‘How the result of a measurement of a component of the spin of a spin- 1 2 particle can turn out to be 100,’ ’’ Physical Review Letters 62, 2325 (1988) (doi: 10.1103/PhysRevLett.62.2325), and Asher Peres, ‘‘Quantum Measurements with Postselection,’’ Physical Review Letters 62, 2326 (1988) (doi: 10.1103/PhysRevLett.62.2326), as well as the reply by Y. Aharonov and L. Vaidman, Physical Review Letters 62, 2327 (1988) (doi: 10.1103/PhysRevLett.62.2327).
19.3 Continuous Position Measurement the sum of the individual measurement strengths, as long as no other transformation or evolution occurs between the two measurements. Notice how the information from the second measurement is incorporated with that of the first. After the first measurement, the best estimate for the position of the quantum system is \alpha, with uncertainty 1/\sqrt\kappa. After the second measurement (where the result is \alpha′), the new best position estimate is an average of the old estimate and the new measurement result,
, (19.56) weighted by the respective uncertainties. The new uncertainty of the estimate is reduced to 1/ \sqrt
19.3 Continuous Position Measurement Now, to construct a continuous measurement of position, we will arrange to have a sequence of weak position measurements, separated in time by ∆t. We will also let the measurement strength depend on time by making the rescaling \kappa −\rightarrow 8\kappa∆t in the measurement operator (19.54), so that
4\kappa∆t \pi 1/4 Z \infty −\infty
(19.57) The factor of 8 here simply gives a convenient normalization for the measurement strength. We will return to the dependence on the time interval ∆t below, but this particular scaling is necessary to obtain a sensible limit as ∆t −\rightarrow 0. Now with this set of measurement operators, the probability of obtaining a particular measurement result \alpha is
= Tr "4\kappa∆t \pi 1/4 Z \infty −\infty
4\kappa∆t \pi 1/4 Z \infty −\infty
= r 4\kappa∆t \pi Z \infty −\infty¶
(19.58) In the limit of small ∆t, the Gaussian factor in the integrand is much broader than the position probability density \langle x|\rho|x\rangle for the quantum state. Since it varies slowly over the scale of \langle x|\rho|x\rangle , the Gaussian factor can be pulled out of the integral, with x replaced by \langle x\rangle , near which \langle x|\rho|x\rangle is peaked. The integral then becomes trivial, and we obtain
r 4\kappa∆t \pi
(19.59) so that the measurement result \alpha is a Gaussian random variable with variance 1/8\kappa∆t. Noting also that
a Gaussian random variable) as
∆W \sqrt 8\kappa∆t (19.60) since the mean and variance agree with those from the probability density (19.59). Recall that \alpha is an index for the measurement operator—equivalently, the measurement result for a particular time interval of duration ∆t—but we may regard it in a sense as a stochastic, dynamical variable, since the measurement is repeated in time.
19.3.1 State Collapse and the Stochastic Schrödinger Equation¶
Chapter 19. Position Measurement 19.3.1 State Collapse and the Stochastic Schrödinger Equation As the stream of measurement results \alpha(t) comes in, the quantum state must be correspondingly modified in light of the new measurement information. Recall that a measurement result of \alpha in a particular time interval of duration ∆t causes the state to transform according to
p
. (19.61) To simplify the evaluation, we can ignore the renormalization factor, so that we can use Eq. (19.57) for the measurement operator to see that the state change is given by
= 4\kappa∆t \pi 1/4 Z \infty −\infty
= 4\kappa∆t \pi 1/4
(19.62) since the x in the exponential is the position operator. Here, the twiddle indicates an unnormalized state vector. Dropping the normalization factor and inserting expression (19.60) for \alpha, we find
" −2\kappa ∆t
∆W \sqrt 8\kappa∆t 2#
∝exp h
\sqrt 2\kappa ∆W i
(19.63) where we have dropped the terms in the exponential that do not involve the position operator. In the infinitesimal limit, we can thus write
h
\sqrt 2\kappa dW i
= h
\sqrt 2\kappa dW
= h 1 −
dt + \sqrt 2\kappa x dW i
(19.64) where we have (without approximation) expanded the exponential to first order in dt and second order in dW, setting dW 2 = dt according to the rules of It¯o calculus. Normalizing the new state vector and expanding to first order in dt (and second in dW),
q
= 1 −
dt + \sqrt 2\kappa x dW
q
\sqrt 2\kappa x dW
= h 1 −
dt −
dW 2 + \sqrt
i
= h
\sqrt
i
(19.65) so that we arrive at the stochastic Schrödinger equation for the continuous position measurement:
\sqrt
(19.66)
19.3.2 Stochastic Master Equation¶
19.3 Continuous Position Measurement Again, accompanying the SSE is the measurement record (19.60), which in the infinitesimal limit becomes
\sqrt 8\kappa, (19.67) where dy := \alpha dt, or directly in terms of \alpha,
\sqrt 8\kappa, (19.68) where as before \xi(t) ≡dW(t)/dt. In terms of \alpha, the variance sensibly diverges in the infinitesimal limit, because the information gained in time dt is zero. To obtain position information, the observer must average dy(t) over some finite time interval: y(t) t = 1 t Z t
\sqrt 8\kappa t Z t dW(t′) = 1 t Z t
\sqrt 8\kappa t (19.69) The second term represents uncertainty in the measurement, and generically converges to zero as 1/ \sqrt t. The first term, which represents the position information, would be simply\langle x\rangle if this expectation value were time- independent, but the integral requires knowledge of the time evolution of the state, and thus its calculation requires the solution to the SSE. 19.3.1.1 Gaussian Noise In constructing the above SSE, we explicitly assumed Gaussian collapse operators Ω(\alpha). This resulted in the Gaussian noise process dW appearing in the SSE, because in the infinitesimal limit (weak-measurement limit), the width and thus also the shape of the collapse operator determined the noise statistics of measurement results \alpha(t). The question is, how general is this? If we had assumed a different form for Ω(\alpha), would we have obtained a different noise process? The answer is that the infinitesimal limit is an idealization, and really we should only consider increments
Z t+∆t t dy(t) (19.70) for the measurement record (and corresponding finite increments for the quantum-state evolution). Such an increment is a sum over arbitrarily many infinitesimal noise increments, and thus under the continuous idealization, any ‘‘reasonable’’ form for the probability distribution of the noise increments will give results equivalent to the choice of Gaussian noise and Gaussian collapse operators, according to the central-limit theorem. By ‘‘reasonable,’’ we first mean that in the above finite construction, before taking the limit ∆t −\rightarrow 0, the variance of the collapse operator (i.e., the variance of the state Ω(\alpha)|\psi\rangle for an initially very uncertain state |\psi\rangle ) should exist and be finite, so that the statistics of many combined collapses are Gaussian. Further, in the limit ∆t −\rightarrow 0, the variance of the infinitesimal increments should be proportional to dt: otherwise, for a variance scaling of the form dt\beta, the above integral ∆y will be either vanish (\beta > 1) or diverge (\beta < 1). Thus, Gaussian noise is general in the sense that any (appropriately normalized) continuous noise process dV representing a continuous measurement may be regarded as the Wiener process dW, so long as the integrated increments ∆V have finite variance. 19.3.2 Stochastic Master Equation As in the case of photodetection, we can generalize the SSE by using it to derive a stochastic master equation, expanding to second order:11
+
. (19.71) 11The Gaussian-projector method in the continuous limit was introduced to derive the unconditioned form for this master equation first by Carlton M. Caves and G. J. Milburn, ‘‘Quantum-mechanical model for continuous position measurements,’’ Physical Review A 36, 5543 (1987) (doi: 10.1103/PhysRevA.36.5543).
19.3.3 Inefficient Detection and Multiple Observers¶
Chapter 19. Position Measurement The resulting SME is
\sqrt
\sqrt
(19.72) where recall the superoperators we defined previously in Chapter 18:
(19.73) Of course, the dW here, while effectively a stochastic variable, is defined in terms of the measurement record (19.67), so that we may eliminate it and write the SME in terms of the measurement results dy:
(19.74) Also, in writing down this SME (as well as the corresponding SSE above), we have ignored any Hamiltonian evolution that proceeds in parallel with the measurement process. Thus, Hamiltonian terms should be added as necessary, so that
\sqrt
(19.75) in the case of the SME with system Hamiltonian H. Of course, the corresponding term may be added to the SSE. 19.3.3 Inefficient Detection and Multiple Observers Notice that we may write the SME (19.72) as
\sqrt
\sqrt
(19.76) and the measurement record as
\sqrt8\kappa1 + dW2 \sqrt8\kappa2 , (19.77) where dW1 and dW2 are Wiener processes, so long as \kappa1 + \kappa2 = \kappa and we thus identify dW = \sqrt\kappa1 dW1 + \sqrt\kappa2 dW2 as the Wiener process from before. We can then associate dW1 and dW2 with different observers, or with the information detected and not detected by a single observer.12 This is precisely the same construction as for photodetection (Section 18.2.8), except now the detector for position information is more abstract.
realizations of dW2 (and then relabel dW1 −\rightarrow dW) to obtain the SME for inefficient detection, where \eta is the fraction of information actually received by the observer:
p
(19.78) Correspondingly, the measurement record becomes
(19.79) 12see also A. Barchielli, ‘‘Stochastic differential equations and a posteriori states in quantum mechanics,’’ Int. J. Theor. Phys. 32, 2221 (1993) (doi: 10.1007/BF00672994); and Jacek Dziarmaga, Diego A. R. Dalvit, Wojciech H. Zurek, ‘‘Conditional quantum dynamics with several observers,’’ Phys. Rev. A 69, 022109 (2004) (doi: 10.1103/PhysRevA.69.022109).
19.3.4 Interpretation¶
19.3 Continuous Position Measurement for the case of inefficient detection. The bad-detection limit \eta = 0 leads to
(19.80) which is simply the unconditioned master equation for a position measurement. In this case, only the disturbance is left, and we will see shortly that the disturbance term here corresponds to momentum diffusion. Similarly, the SME (19.76) represents the evolution for position measurement by two observers that do not share information, but from the point of view of an omniscient observer. Observers 1 and 2 thus have their own SMEs, given by tracing out the other noise process,
\sqrt
\sqrt
- (19.81) Of course, the disturbance for both measurement processes are present, independent of the ensemble averages. The corresponding measurement records for each observer are
\sqrt8\kappa1
\sqrt8\kappa2 . (19.82) The Wiener processes dW ′ 1,2 for the individual observers are in general different from the corresponding processes dW1,2 from the omniscient observer, because in the individual master equations, the expectation values are taken with respect to \rho1,2 rather than \rho. Equating (\kappa1 dy1 + \kappa2 dy2)/\kappa with dy, \kappa1
\sqrt 8 \kappa +
\sqrt 8 \kappa
\sqrt 8\kappa, (19.83) we can split up the two terms on the right-hand side consistently and separate the parts depending on \kappa1 and \kappa2, so that for example \kappa1
\sqrt 8 \kappa = \kappa1
r\kappa1 \kappa dW \sqrt 8\kappa, (19.84) and find dW ′ 1 = \sqrt
dW ′ 2 = \sqrt
(19.85) for the relation between the noise sources of the individual observers to the noise sources of the omniscient observer. 19.3.4 Interpretation To better see the effects of the measurement terms in the SME (19.72), we will look at the evolution of the first- and second-order moments, as we did in the case of photodetection. Again deriving the equation of motion for the expectation value of an arbitrary operator A, essentially by taking Eq. (18.131) and setting a −\rightarrow x, we obtain (now including Hamiltonian evolution)
p
h
i dW. (19.86) Assuming for simplicity a Hamiltonian evolution according to the harmonic-oscillator Hamiltonian H = p2 2m + 1 2m\omega 2x2, (19.87)
Chapter 19. Position Measurement it follows that the means and variances obey the evolution equations13
p
p
x
xp
mVp −m\omega 2 0 Vx −8\eta\kappaVxCxp, (19.88) where again the variances Vx and Vp are defined by V\alpha :=
\alpha2 −\langle \alpha\rangle 2, and the symmetrized covariance is
Gaussian state, for which the moments obey [see Eqs. (18.136)]
x3
[x, p2]+
h
h
. (19.89) These relations explicitly decouple the means and variances from the higher-order moments. Also, since the Gaussian state is preserved both by the harmonic Hamiltonian evolution and by the position measurement (which amounts to a Gaussian collapse at each instant in time), there is no loss of generality involved in assuming a Gaussian state provided that the system starts in a Gaussian state, and even if the system starts in some other state, the position measurement will eventually force the system into a Gaussian state. In examining Eqs. (19.88), we can simply use the coefficients to identify the source and thus the interpretation of each term. The first term in each equation is due to the natural Hamiltonian evolution of the harmonic oscillator. Terms originating from the D[c]\rho component are proportional to \kappa dt but not \eta; in fact, the only manifestation of this term is the 2¯h2\kappa term in the equation of motion for Vp. Thus, a position measurement with rate constant k produces momentum diffusion (heating) at a rate 2¯h2\kappa, as is required to maintain the uncertainty principle as the position uncertainty contracts due to the measurement. (This can also be seen by deriving the Fokker–Planck equation for the Wigner function; see Problem problem:fokker- planck-x-measurement.) There are more terms here originating from the H[c]\rho component of the master equation, and they are identifiable since they are proportional to either \sqrt\eta\kappa or \eta\kappa. The dW terms in the equations for \langle x\rangle and \langle p\rangle represent the stochastic nature of the position measurement. That is, during each small time interval, the wave function collapses slightly, but we don’t know exactly where it collapses to. The stochastic term in the \langle x\rangle equation is proportional to Vx, since the larger the variance, the wider the range of potential collapses. The stochastic term in the \langle p\rangle equation is proportional to Cxp, since a position measurement only induces
behavior that we saw in Eq. (19.60). The more subtle point here lies with the nonstochastic terms proportional to \eta\kappa, which came from the second-order term, as in the last term of dVx = d
X2
(19.90) where It¯o calculus generates a nonstochastic term from dW 2 = dt. Notice in particular the term of this form in the Vx equation, which acts as a damping term for Vx. This term represents the certainty gained via the measurement process. The other similar terms are less clear in their interpretation, but they are 13A. C. Doherty and K. Jacobs, ‘‘Feedback control of quantum systems using continuous state estimation,’’ Physical Review A 60, 2700 (1999) (doi: 10.1103/PhysRevA.60.2700).
19.3.5 Linear Stochastic Evolution Equations¶
19.3 Continuous Position Measurement necessary to maintain consistency of the evolution. Essentially, again Vp and Cxp are only modified by a position measurement, which you might expect would only influence Vx, if x and p are correlated. Hence the presence of Cxp in these terms. Note that we have made the assumption of a Gaussian initial state in deriving these equations, but this assumption is not very restrictive. Due to the linear potential and the Gaussian collapse operators, these equations of motion preserve the Gaussian form of the initial state. The Gaussian collapses additionally converts arbitrary initial states into Gaussian states at long times. Furthermore, as we have mentioned, the assumption of a Gaussian measurement is not restrictive—under the assumption of sufficiently high noise bandwidth, the central-limit theorem guarantees that temporal coarse-graining yields Gaussian noise for any measurement process giving random deviates with bounded variance. As a simple example to illustrate the conditioned dynamics, below is plotted the evolution of a free
particle. In the measured case, the initial variance Vx = 2 contracts due to the measurement, because position becomes more certain under the measurement process, until dispersion and the measurement balance. The centroid also moves stochastically due to the random nature of the measurement process. Of course, the stochastic motion is different for each possible realization of the measurement process. x \leftarrow time x 19.3.5 Linear Stochastic Evolution Equations Recall from Section 19.1.2 that the infinitesimal transformation \rho −\rightarrow Ω(dW)\rhoΩ\dagger(dW), with collapse oper- ator
¯h dt −1 2c\daggerc dt + c dW, (19.91) leads to the unnormalized evolution equation
dW, (19.92)
Chapter 19. Position Measurement where ˜\rho is the unnormalized density operator. Adding the nonlinear term −
\rho dW restores the proper trace of \rho, and thus leads to the familiar normalized SME
dW −
\rho dW = −i
(19.93) This linear SME is clearly equivalent to the linear SSE
(19.94) where | ˜\psi\rangle is the unnormalized state vector, since this corresponds to the infinitesimal transformation | ˜\psi\rangle −\rightarrow Ω(dW)| ˜\psi\rangle . The corresponding normalized SSE reads
c\daggerc −
c + 1
c −1
(19.95) as we saw before for homodyne detection, Eq. (18.74). We may regard the corresponding measurement record (up to an arbitrary factor) in either case to be
dt + dW. (19.96) Despite the fact that it does not preserve the norm of the state, the linear SME (19.92) and linear SSE (19.94) are still useful—provided that they are interpreted properly—precisely because they are linear and thus facilitate analytic solutions.14 19.3.5.1 Norm of the Linear Solution To see the effect of using the unnormalized equation, consider the evolution of the norm of the state in an infinitesimal time interval, assuming the state is initially normalized:
(19.97) That is, the norm is just the probability that the outcome labeled by the value of dW occurred. The normaliza- tion factors for evolution in subsequent time intervals simply multiply, and so for an initially normalized state
dW(t) of the measurement actually occurred. 19.3.5.2 Interpretation of the Solution However, for the same realization dW(t), the evolution according to the linear SSE (SME) is not equivalent to evolution according to the normalized SSE (SME). We can see this because the linear evolution is given in terms of the transformation operator Ω(dW) (19.91), which is labeled solely by the stochastic variable dW, which is independent of the state. By contrast, the normalized evolution of the state corresponds to collapse operators Ω(dr), which are labeled by the measurement result dr(t) (19.96), which depends on the state via
measurement results by the amount
. Thus, the unnormalized evolution according to the linear SSE (SME), after renormalization at the end, corresponds to the correct normalized evolution for some possible measurement realization dr(t), just not the one given by Eq. (19.96). In fact, the measurement record corresponding to the linear evolution is generically much less likely than, say, the realization according to Eq. (19.96), which occurs with the same probability as that for choosing the realization of dW(t). However, 14Peter Goetsch and Robert Graham, ‘‘Linear stochastic wave equations for continuously measured quantum systems,’’ Physical Review A 50, 5242 (1994) (doi: 10.1103/PhysRevA.50.5242); H. M. Wiseman, ‘‘Quantum trajectories and quantum measurement theory,’’ Quantum and Semiclassical Optics 8, 205 (1996) (doi: 10.1088/1355-5111/8/1/015).
19.3 Continuous Position Measurement¶
actually occurred, and so as long as we weight these trajectories appropriately in any ensemble average, we have no problem. Thus, to summarize: linear SSEs and SMEs give different conditioned evolution than their normalized counterparts, and in fact they realize particular evolutions with the wrong probability (assuming dW(t) is chosen with the proper probability), but the probability for realizing any particular simulated trajectory is given by the final norm of the state. Simulation according to the normalized equations always gives solutions that are equally likely if dW(t) is chosen with the proper probability. To see another example of this in terms of the position measurement, recall from Eq. (19.64) that the infinitesimal evolution under a continuous position measurement is given by
h 1 −
dt + \sqrt 2\kappa x dW i
(19.98) where, although the result is not normalized, the result is chosen with the correct probability by our con- struction. The corresponding position measurement record was
\sqrt 8\kappa, (19.99) in terms of which we can write the above evolution as
(19.100) Now if we make the replacement dy −\rightarrow dW/ \sqrt 8\kappa, that is we shift the mean of the distribution of position- measurement results, we obtain the infinitesimal evolution
h
\sqrt 2\kappa x dW i
(19.101) corresponding to the linear SSE for position measurement,
\sqrt
(19.102) or equivalently the SME
\sqrt
(19.103)
\sqrt 2\kappa x. 19.3.5.3 Explicit Solutions of Measurement Dynamics In the linear form, then, analytic solutions become more tractable. The solution to the linear SSE (19.94) is thus given by noting that
a + b2 dt + b dW, (19.104) so that composing infinitesimal exponential evolution operations gives
−i ¯h Z t H(t′) dt′ −1 Z t
dt′ + Z t c dW(t′)
(19.105) if c and H commute, which in the case of having no explicit time dependence of c (i.e., the measurement is not changed with time) and a time-independent Hamiltonian becomes
−iHt ¯h −1
t + c W(t)
(19.106)
Chapter 19. Position Measurement Recall that W(t) is a Gaussian random variable with zero mean and variance t, so that we do not need the entire history dW(t) to compute the final solution. Now consider a nondemolition measurement of a Hermitian observable operator Q (i.e., the mea-
\sqrt 2\kappaQ in the above equations, and have a corresponding measurement record
\sqrt 8\kappa (19.107) for the conditioned evolution. Then the final state is
−iHt ¯h
\sqrt 2\kappa Q W(t)
(19.108) Decomposing the initial state into eigenstates |q\rangle of Q,
X q cq|q\rangle , (19.109) we may write the final state as
X q cq exp −iEqt ¯h
\sqrt 2\kappa q W(t) |q\rangle , (19.110) where H|q\rangle = Eq|q\rangle . Now recall that the probability of this particular outcome is given by the norm of the final state
X q |cq|2 exp h
\sqrt 8\kappa q W(t) i . (19.111) or in terms of the rescaled Wiener process
\sqrt 8\kappa t, (19.112) the norm becomes
X q |cq|2 exp
. (19.113) But the actual probability for realizing Y (t) is the probability for picking Y (t) in a simulation (i.e., the corresponding Gaussian probability for picking W(t)), multiplied by the norm of the final state:
\sqrt
= r 4\kappat
q |cq|2 exp
= r 4\kappat \pi X q |cq|2e−4\kappat(Y −q)2. (19.114) That is, the probability distribution for Y (t) is a sum of Gaussians of width 1/ \sqrt 8\kappat, centered about each eigenvalue q and weighted by the usual Born probability c 2 q . But recalling that using the linear SSE amounts to choosing the measurement record with the expectation value of the observable removed, dy = dW \sqrt 8\kappa, (19.115)
19.4.1 Center-of-Mass Dynamics¶
19.4 Imaged Resonance Fluorescence as a Position Measurement and thus we interpret
t Z t dy(t′) (19.116) as simply the time-averaged measurement result. Thus, we have shown in the case where there is no real interplay between the measurement and the Hamiltonian evolution, we can solve the measurement evolution explicitly and see that the time average of the measurement record gives the observed value of Q for a particular trajectory (particular experiment). This observed value converges (almost always) to an eigenvalue of the discrete observable as 1/ \sqrt t. 19.4 Imaged Resonance Fluorescence as a Position Measurement Now we will consider several physical examples of position measurements, taking a single atom as a concrete example of a quantum-mechanical particle. The first one we will consider is the case of photodetection of ordinary resonance fluorescence, where the atom is excited by a plane wave, as we have already treated in Chapter 18. However, to gain position information, it is not sufficient to simply detect the photons, you have to image the scattered light, just as you would use a camera or a microscope to locate a small object. 19.4.1 Center-of-Mass Dynamics We can write the SME for spontaneous emission as usual as
¯h[H, \rho]dt −\Gamma
(19.117) with superoperators
\rho, (19.118) and a Poisson process characterized by
dt. (19.119) Now we want to consider how the evolution of the atomic internal state influences the atomic center-of- mass motion. We need to explicitly include the mechanical effects of the resonance fluorescence. First, we will model the situation of angle-resolved photodetection, where we break up the Poisson process dN into many infinitesimal Poisson processes dN(\theta, \phi)/dΩ, corresponding to emission in any possible direction (\theta, \phi). Additionally, if the photon is detected in the direction k, then the atom must recoil with momentum −¯hk, which is equivalent to applying the operator
(19.120) to the atomic state at each detection event, rather than just the lowering operator \sigma. Thus, for the term describing each subprocess dN(\theta, \phi)/dΩ, we make the replacement
(19.121) where k points along the direction (\theta, \phi), and then sum over all angles to obtain
¯h[H, \rho] dt −\Gamma
Z dΩJ
dΩ . (19.122) (Compare to the form (5.404) of the atom–field interaction Hamiltonian, where the spatial dependence of the field enters via the Rabi frequency.) The Poisson processes are characterized by the means
dΩ
(19.123)
19.4.2 Imaging¶
Chapter 19. Position Measurement where fˆ\epsilon(\theta, \phi) is the dipole radiation pattern for a dipole unit vector of ˆ\epsilon. In the Weisskopf–Wigner treatment of spontaneous emission in Chapter 11, we obtained this master equation by simply accounting for the spatial dependence of the vacuum field modes by letting gk −\rightarrow gkeik\cdotr (19.124) in the atom–field interaction Hamiltonian to obtain HAF = X k,\zeta ¯h
, (19.125) and then expanding the state vector also in the momentum basis
Z
Z d3p X k,\zeta
(19.126) In the weak-excitation limit, we can take the magnitude of k to have the value of an externally applied probe field, which will generally be near enough to resonance that it will match the resonant wave number (if the detuning is very large, the Rayleigh-scattered photons are elastically scattered from the incident field and thus have the same wave number as the driving field). Recall that the measurement terms only account for the momentum recoil on emission; any additional recoil due to photon absorption is already accounted for by the Hamiltonian evolution (see, e.g., Section 5.8.6.6). We can simplify the angle-resolved SME by carrying out the angular integral, defining dN to be one
¯h[H, \rho]dt −\Gamma
\rho dN, (19.127) with
dt (19.128) as before. The angles \theta and \phi are then stochastic variables with probability density f(\theta, \phi) sin \theta. 19.4.2 Imaging The above master equation (19.122) is for an angle-resolving detector. What we see is that angle-resolved detection keeps explicit track of the atomic momentum kicks due to spontaneous emission. An imaging detector, on the other hand, gives up resolution of the direction of the emitted photon wave vector k, thus obtaining instead some position information about the atom. An imaging system operates by summing fields from many directions together and then detecting the resulting interference pattern. The procedure for obtaining the measurement operators for the imaging system is as follows.15 Notice that we can regard the master equation (19.122) as a normal jump process of the form (19.117), with measurement operators
p
(19.129) where we sum over all possible emission angles. In writing down this family of operators, we are specializing
k. This operator ranges from −1 to 1 in cos \theta and from 0 to 2\pi in \phi. Thus, we can write down Fourier coefficients (operators) for \sigma(\theta, \phi), since these functions are defined on a bounded domain, with two indices
\sigma \sqrt 4\pi Z 2\pi d\phi Z 1 −1 d(cos \theta) p
(19.130) 15M. Holland, S. Marksteiner, P. Marte, and P. Zoller, ‘‘Measurement Induced Localization from Spontaneous Decay,’’ Phys- ical Review Letters 76, 3683 (1996) (doi: 10.1103/PhysRevLett.76.3683); W. Greenwood, P. Pax, and P. Meystre, ‘‘Atomic transport on one-dimensional optical lattices,’’ Physical Review A 56, 2109 (1997) (doi: 10.1103/PhysRevA.56.2109). For one experimental implementation, see Hidetoshi Katori, Stefan Schlipf, and Herbert Walther, ‘‘Anomalous Dynamics of a Single Ion in an Optical Lattice,’’ Physical Review Letters 79, 2221 (1997) (doi: 10.1103/PhysRevLett.79.2221).
19.4 Imaged Resonance Fluorescence as a Position Measurement However, this expression corresponds to imaging via an ideal imaging system, where the aperture extends over the full 4\pi solid angle (requiring, for example, arbitrarily large lenses on either side of the atom). In practice it is rare to come anywhere close to this extreme. Thus, we include the effects of an aperture that only allows the imaging system to detect radiated light within a limited solid angle. We thus take the intensity transmission of the aperture to be represented by the function T(\theta, \phi), so that we explicitly ignore any phase-shift effects of the aperture. The collapse operator for imaged detection is thus modified to be the Fourier transform of the angular distribution, which now includes the aperture function:
\sigma \sqrt 4\pi Z 2\pi d\phi Z 1 −1 d(cos \theta) p
(19.131) Of course, any phase mask could be modeled by introducing a factor exp[iϕ(\theta, \phi)] in the above integrand. Now consider the \phi part of the above integral, which we may write as
\sqrt 2\pi Z 2\pi d\phi p
(19.132) Note that in this part of the measurement operator, there is no position dependence, and thus we will be able to eliminate it from the dynamics. With our normalization convention, we have chosen our normalized basis functions as ei\beta\phi/ \sqrt 2\pi, and thus \infty X
(19.133) where the argument of the delta function is taken modulo 2\pi, so that we have the overall normalization \infty X
a\dagger
Z 2\pi
(19.134) If we trace over the measurement result \beta, this amounts to using the reduced density operator \infty X
Z 2\pi
(19.135) which is equivalent to making the replacement a\beta −\rightarrow q ˜T(\theta) (19.136) in the operator ˜\sigma\alpha\beta, where we now have the effective aperture
Z 2\pi
(19.137) Thus, the measurement operator (19.131) loses the irrelevant index \beta, and reduces to
(19.138) where the effect on the position degree of freedom due to the photodetection is given by the operator
N \sqrt Z 1 −1 d(cos \theta) q ˜T(\theta) e−ikz cos \theta, (19.139) and N is a normalization constant that does not influence the effect of the operator. The A(z) operators, for ‘‘reasonable’’ apertures, contain localized functions of the position z, and thus correspond to position
Chapter 19. Position Measurement measurements. Again, the effect of the operator ˜\sigma\alpha is equivalent to that of the original form ˜\sigma\alpha\beta, but with a trace over \beta:
\infty X
(19.140) The idea here is that for motion along the z-axis, photons going into any azimuthal angle \phi are equivalent as far as providing position information about the atom. Thus, the \theta dependence of the aperture will be most important, but the \phi dependence gives some effective \theta dependence if the aperture is not separable in
Notice that with the normalization convention for the Fourier coefficients here, if we remove the
Z
X \alpha
(19.141) so that the set of measurement operators is complete and properly normalized in either basis. An arbitrary aperture mask will then reduce the efficiency of the measurement, since not all of the photons will be detected. In this case, X \alpha
(19.142) where we have defined the detection efficiency of the angular aperture by
Z 2\pi d\phi Z 1 −1
Z 1 −1
(19.143) While this is the efficiency for photon detection, we will see that in general this is not the same as the efficiency for information gain. If we then choose the normalization N = \sqrt\etaΩ, we will thus have the detection operator
(19.144) where the effect on the position degree of freedom due to the photodetection is given by the operator
Z 1 −1 d(cos \theta) q ˜T(\theta) e−ikz cos \theta, (19.145) so that we associate the efficiency of the detection explicitly with the collapse operator, and now in view of the normalization (19.142), the operators A(z) form a POVM. We can thus get the imaged-detection SME from the angle-resolved form (19.122) by first separating the angular part of the measurement term according to what fraction of light at a given angle makes it through the aperture:
¯h[H, \rho]dt −\Gamma
Z dΩJ \sigmae−ikz cos \theta
dΩ + Z dΩJ \sigmae−ikz cos \theta
dΩ . (19.146) Here, dN1(\theta, \phi)/dΩenumerates the processes by which a photon is detected through the aperture, satisfying
dΩ
(19.147) and dN2(\theta, \phi)/dΩrepresents the fictitious processes by which the photons that are blocked by the aperture are detected, and thus satisfies
dΩ
(19.148)
19.4 Imaged Resonance Fluorescence as a Position Measurement so that taking the two processes together is equivalent to the original model. Eliminating the undetected photons, we take an ensemble average over dN2/dΩto obtain
Z 1 −1
\sigmae−ikz cos \theta
\Gamma
Z dΩJ \sigmae−ikz cos \theta
dΩ , (19.149) where the second term is the quantum backaction due to the undetected photons, the third term is the backaction due to the detected photons, and the last term represents the gain of measurement information. Here we have defined the angular distribution of blocked photons
Z 2\pi
(19.150) so that the transmitted and blocked distributions add to the ‘‘natural’’ radiation pattern when there is no aperture:
Z 2\pi
(19.151) Now from our argument relating the position-sensitive operators ˜\sigma\alpha to the momentum-kicking operators \sigmae−ikz cos \theta, we may rewrite the last term as a sum over \sigma\alpha:
Z 1 −1
\sigmae−ikz cos \theta
\Gamma
\infty X
(19.152) This amounts to a unitary transformation on the measurement operators, under which the rest of the master equation is invariant, as implied by the completeness in Eq. (19.140). Here, the Poisson processes are characterized by
dt, (19.153) so that the probability of the measurement outcome \alpha goes as the squared modulus of the overlap of A(z −\alpha\lambda/2) with the center-of-mass part of the atomic wave function \psi(z). Again, we may combine the Poisson processes dN\alpha into a single process dNΩ,
Z 1 −1
\sigmae−ikz cos \theta
\Gamma
(19.154) where the combined Poisson process is characterized by
dt, (19.155) and the probability distribution for the integer outcome \alpha is the expectation value
Z
(19.156) The expectation value is taken with respect to the excited-state part |e\rangle of the atomic state, since this is what survives after the reduction \rho −\rightarrow \sigma\rho\sigma\dagger/
representing the photon detection. Notice also that the set of possible measurement values is not continuous, but rather is discretely spaced by \lambda/2, which is rather odd for the result of a position measurement. However, this is the case because the Fourier transform was taken with respect to a bounded domain, and is thus a consequence of the sampling theorem (Chapter 25): the information contained in a function on a bounded, continuous domain is equivalent in the Fourier domain to the information in a function on a discrete (but infinite) domain.
Chapter 19. Position Measurement 19.4.2.1 Example: 4\pi Detection As an example of a particular form for the position-reduction operator A(z), a radiating atomic dipole oriented along the z-axis has
(19.157) in which case the effective aperture becomes
4 sin2 \theta. (19.158) This gives the measurement operator
r 3\pi2 J1(kz) kz , (19.159) where J1(x) is an ordinary Bessel function. Since J1(x) decays as x−1/2 for large x, the tails of the collapse operator here decay as z−3/2. 19.4.2.2 Example: Small Gaussian Aperture Often in real situations, the aperture subtends only a small solid angle. Intuitively, one expects a camera imaging system to be most effective when oriented normal to the z-axis, so we choose the aperture to be centered about \theta = \pi/2. detector dq x y z We can also arbitrarily take the aperture to be centered about \phi = 0. It is mathematically convenient to assume a Gaussian transmission function for the aperture, and we thus take the intensity transmission function of the aperture to be
(\delta\theta)2 exp −2\phi2 (\delta\phi)2 (19.160) where \delta\theta and \delta\phi are the same for a circular aperture. Now the effective aperture becomes
(\delta\theta)2 , (19.161) where the azimuthal angular integral is
Z \pi −\pi d\phi exp −2\phi2 (\delta\phi)2
(19.162) We can suppress the dependence on \theta by assuming that the distribution function f(\theta, \phi) varies slowly over the width \delta\theta of the aperture (in particular, we should not choose the aperture to be near a direction in which the radiation pattern vanishes). This happens when the aperture is narrow, and the result is the constant efficiency
Z \pi −\pi d\phi exp −2\phi2 (\delta\phi)2
(19.163)
19.4.3 Adiabatic Elimination of the Internal Atomic State¶
19.4 Imaged Resonance Fluorescence as a Position Measurement¶
\eta\phi \approx 3 8\pi Z \infty −\infty d\phi exp −2\phi2 (\delta\phi)2 =
\sqrt 2\pi (19.164) in the limit where \delta\phi is small. If \delta\theta is small, then the integrand is only appreciable for \theta near \pi/2 due to the Gaussian factor. Recentering the integrand in Eq. (19.145), making the small-angle approximation in the rest of the integrand, and extending the limits of integration, we find
2\etaΩ Z \pi/2 −\pi/2 d\theta cos2 \theta e−ikz sin \theta exp − \theta2 (\delta\theta)2 \approx
2\etaΩ Z \infty −\infty d\theta e−ikz\theta exp − \theta2 (\delta\theta)2 =
2\etaΩ
" −
2 z2 # . (19.165) Thus, the measurement operator in this case is also Gaussian. We can write the fraction of photons trans- mitted by the aperture as
Z 1 −1
r\pi
(19.166) in the same regime of small \delta\theta, and thus the Gaussian operator becomes
sr\pi
" −
2 z2 # (19.167) upon eliminating \etaΩ. 19.4.3 Adiabatic Elimination of the Internal Atomic State So far, we have seen how the internal and external dynamics of the atom are intrinsically linked. Now we want to focus on the external atomic dynamics. To do so, we will take advantage of the natural separation of the time scales of the dynamics. The internal dynamics are damped at the decay rate \Gamma, which is typically on the order of ∼107 s−1. The external dynamics are typically much slower, corresponding to kHz or smaller oscillation frequencies for typical laser dipole traps. The adiabatic approximation assumes that the internal dynamics equilibrate rapidly compared to the external dynamics, and are thus always in a quasi-equilibrium state with respect to the external state. 19.4.3.1 Internal Quasi-Equilibrium At this point, we must consider the internal atomic dynamics more precisely, and as a review we will compactly rederive the necessary steady-state results from Section 5.5.1. A resonant, driving (classical) laser field enters in the usual form ˜HAF = ¯hΩ
, (19.168) where the Rabi frequency Ωcharacterizes the strength of the laser–atom interaction, and we have included the
our attention again to the z-axis, the Hamiltonian becomes ˜HAF = ¯hΩ
, (19.169)
Chapter 19. Position Measurement In writing down this interaction, we have made (as before) the standard unitary transformation to a rotating frame where the free atomic Hamiltonian ˜HA = 0. Note that if the driving field propagates along a normal to the z-axis, the spatial dependence of the field vanishes in ˜HAF. The usual unconditioned master equation with this interaction, but neglecting the external motion (that is equivalent to the usual, on-resonance optical Bloch equations) is
(19.170) This equation implies that the expectation value of an operator A evolves as
¯h D [A, ˜HD] E + \Gamma
. (19.171) This gives the following equations of motion for the density-matrix elements \rho\alpha\beta := \langle \alpha|\rho|\beta\rangle :
= iΩ h
\sigma\dagger eikDz cos \thetaDi −\Gamma
,
−
−\Gamma
(19.172) The remaining matrix elements are determined by \rhoge = \rho∗ eg and \rhogg =
= 1 −
. Setting the time derivatives to zero, we can solve these equations to obtain (as we already derived in Section 5.5.1)
t\rightarrow \infty= Ω2/\Gamma2
, (19.173) for the internal steady-state of the atom. 19.4.3.2 External Master Equation To make the adiabatic approximation and eliminate the internal dynamics, we note that there is no effect on the external dynamics apart from the slow center-of-mass motion in the potential V (x) and the collapses due to the detection events. When the internal timescales damp much more quickly than the external time scales, we can make the replacements
−\rightarrow
t\rightarrow \infty
(19.174) in the master equation (19.154) and mean Poisson process (19.155). In this approximation, we will similarly ignore the fast fluctuations of the atomic operators, which do not substantially couple to the slow atomic dynamics, and thus also make the replacements
t\rightarrow \infty
(19.175)
\approx
\sigma\dagger
\Gamma2 . (19.176) In this case, the master equation simplifies and becomes
Z 1 −1
eikDz cos \thetaDe−ikz cos \theta
\rho dNΩ, (19.177)
19.4.4 White-Noise Limit: Gaussian Aperture¶
19.4 Imaged Resonance Fluorescence as a Position Measurement where the mean of the Poisson process becomes
(19.178) and we have defined the mean spontaneous-scattering rate
. (19.179) The Hamiltonian HCM refers to the external, center-of-mass Hamiltonian for the atom, since the evolutions according to the atomic Hamiltonian ˜HA and ˜HAF are trivial in this regime. Notice that the disturbance (backaction) term contains two momentum-shift operators, a deterministic one for absorption and a random one for spontaneous emission. The absorption disturbance can be eliminated in this one-dimensional analysis
There is effectively now no dependence on the internal atomic degrees of freedom in the master equation, since all such dependence has been reduced to constant averages. We can thus take a partial trace over the internal degrees of freedom by defining the external density operator
(19.180) so that we obtain
Z 1 −1
eikDz cos \thetaDe−ikz cos \theta \rhoext dt + J
\rhoext dNΩ. (19.181) Now we have what we want: a master equation for the atomic center-of-mass state that exhibits localizing collapses due to a physical measurement process. What we essentially have is continuous evolution, with the end of each interval of mean length (\etaΩ\gamma)−1 punctuated by a measurement reduction of the form
. (19.182) But note that here there is extra disturbance for the amount of information we gain, because the aperture only picks up a fraction of the available information. We will return to this point shortly. 19.4.4 White-Noise Limit: Gaussian Aperture Now we will take the white-noise limit, and we will thus obtain a master equation in the standard form for an inefficient, continuous position measurement. To do this, we will consider the case of a small Gaussian aperture, for which we showed the collapse operator was Gaussian and given by Eq. (19.167). As in the finite- step construction of the continuous position measurement, the Gaussian collapse operator A(\alpha) is applied
of-mass motion, the collapses come quickly compared to the motion. Then it is a good approximation (in the temporal coarse-graining sense) to take the formal limit ∆t −\rightarrow 0, while keeping the rate of information gain constant. 19.4.4.1 Spatial Continuum Approximation If an atom is initially completely delocalized, after one photon is detected and the collapse operator A(z−\alpha′) applies, where \alpha′ = \alpha\lambda/2, the atom is reduced to a width of order
\lambda
(19.183) Since this is much larger than the spacing
2 , (19.184)
Chapter 19. Position Measurement it is effectively impossible to ‘‘see’’ the discreteness of the measurement record, and it is a good approximation to replace the set of measurement operators with a set corresponding to a continuous range of possible measurement outcomes. Since in the limit of small spacing ∆x, it is a good approximation to write an integral as a sum X n
Z dx f(x) (19.185) for an arbitrary function f(x), we can make the formal replacement
\sqrt ∆\alpha (19.186) to obtain the continuum limit of the position collapse operators with proper normalization, now regarding \alpha′ as a continuous position index rather than a discrete real index of spacing \lambda/2. Dropping the prime from \alpha′, and using Eq. (19.183) in Eq. (19.167), we now have the collapse operator
s \sqrt
−(z −\alpha)2
. (19.187) We have implicitly written down this operator in the position basis, so technically we should write
Z
s \sqrt
−(z −\alpha)2
(19.188) to be general. Again, \alpha is now a continuous index with dimensions of length, rather than an integer index.
pump beam to be orthogonal to the z-axis), we now have
Z 1 −1
e−ikz cos \theta \rhoext dt + J [A(z −\alpha)]\rhoext dNΩ. (19.189) The probability density for \alpha is then, with the appropriate modification of Eq. (19.156),
Z
Z
\sqrt
−(z −\alpha)2
. (19.190) If the atomic wave packet is well localized beyond the scale \delta\alpha, the probability distribution is thus Gaussian with variance (\delta\alpha)2. 19.4.4.2 Quantum-State Diffusion Comparing the collapse operator A(z) of Eq. (19.188) with the collapse operator (19.57) we see that they are the same if we identify
(19.191) Solving for the measurement strength \kappa, using ∆t = 1/\etaΩ\gamma,
2\lambda2 . (19.192) Repeating the procedure of Section 19.3.1, we can take the limit ∆t −\rightarrow 0 with \kappa fixed. Here, however, this is a mathematical, coarse-graining approximation, as the measurements are really still occuring with a nonzero mean time between collapses. The resulting master equation, in ‘‘quantum-state diffusion’’ form, is
Z 1 −1 d(cos \theta) ˜R(\theta) D[e−ikz cos \theta]\rhoext dt
\sqrt 2\kappaH[z]\rhoext dW. (19.193)
19.4 Imaged Resonance Fluorescence as a Position Measurement The form here is the same as in Eq. (19.72), except for an extra ‘‘disturbance term’’ representing the undetected photons. 19.4.4.3 Diffusion Rates To simplify the master equation (19.193), we will analyze the diffusion rates due to the second and third terms (proportional to \gamma and \kappa, respectively). From the analysis of Eqs. (19.88), recall that the term 2\kappaD[z]\rhoext dt causes diffusion in momentum at the rate
(19.194)
below. We can compute the total diffusion rate due to the spontaneously emitted photons as follows. Each photon emission causes a momentum kick of magnitude ¯hk cos \theta, and the spontaneous emission rate is \gamma. Averaging over the angular photon distribution (19.157), the diffusion rate becomes
Z
. (19.195) On the other hand, we can compute the diffusion rate due only to the detected photons. Using the Gaussian aperture function (19.161) with Eq. (19.164) for the azimuthal part (assuming the usual dipole-radiation pattern) to obtain
\delta\theta r \pi exp
(\delta\theta)2 , (19.196) the partial diffusion rate for detected photons is
Z 1 −1
\delta\theta r \pi Z 1 −1 d(cos \theta) cos2 \theta exp
(\delta\theta)2 \approx \etaΩ
(19.197) where we again used the fact that \delta\theta is small. This is precisely the same rate as D\kappa, since they are two different representations of the same physical process. We see now that the second and third terms of Eq. (19.193) have the same effect of momentum diffusion, both corresponding to heating from photon scattering, but at different rates, corresponding to the partition between detected and undetected photons. We can combine them to obtain
¯h[HCM, \rhoext]dt + 2keffD[z]\rhoext dt + p 2\etaeffkeffH[z]\rhoext dW, (19.198) where the effective measurement strength is keff = DSE
10 , (19.199) and the effective measurement efficiency is
\kappa keff = 5
(19.200) Notice that since \delta\theta is assumed small, the apparent efficiency \etaeff derived from comparing the information rate to the disturbance rate, is much smaller than the photon-detection efficiency of \etaΩ. Evidently, the photons radiated near \theta = \pi/2 are much less effective compared to the photons radiated near \theta = 0 or \pi. This result is counterintuitive when considering typical imaging setups as we have considered here, but suggests that other ways of processing the radiated photons (e.g., measuring the phase of photons radiated closer to the z-axis) are more effective than camera-like imaging.
19.5.1 Localized Probe Field¶
Chapter 19. Position Measurement 19.5 Position Measurement via Excitation by a Local Probe Field Now we will examine a position-measurement method for an atom that uses resonance fluorescence but uses the resolution of a focused probe beam instead of the resolution of an imaging system to gain the position information. This setup is cleaner in the sense that the efficiency is only determined by the photon detection probability, and the form and width of the collapse operator can be chosen essentially independently of the measurement efficiency. 19.5.1 Localized Probe Field Recall from Eq. (19.122) that the quantum-jump SME for resonance fluorescence from a two-level atom is given by
¯h[ ˜H, \rho]dt −\Gamma
Z dΩJ
dΩ . (19.201) in the case of angle-resolved detection of the photons, where again the Poisson processes corresponding to each angular element are
dΩ
(19.202) The Hamiltonian describes both the atomic motion and the atom–field coupling, so we can decompose it into these parts as ˜H = HCM + ˜HAF, (19.203) where the center-of-mass Hamiltonian describes one-dimensional motion along the z-axis, H = p 2 z 2mA + V (z), (19.204) where V (z) is some external atomic potential, and the atom–field coupling Hamiltonian is given in the rotating frame of the laser field from Eq. (5.406) by ˜HAF = ¯h
, (19.205) where Ω(z) is the space-dependent Rabi frequency representing the resonant probe field, defined such that |Ω(z)|2 is proportional to the local intensity. We assume the probe to propagate normal to the z-axis, so that we can assume a zero average momentum recoil on absorption. In the weak-excitation regime, where the rate of spontaneous emission is much smaller than the excited- state decay rate \Gamma, we can adiabatically eliminate the atomic internal state, which as in Section (19.4.3)
lowest order in Ω/\Gamma). With this replacement, the SME (19.201) becomes
¯h[HCM, \rho]dt −1 2\GammaH |Ω(z)|2
Z dΩJ Ω(z)e−ik\cdotr
dΩ . (19.206) while the mean of the Poisson process reduces to
\Gamma
|Ω(z)|2 dt. (19.207) Clearly, now, the electric-field profile Ω(z) of the probe acts as a collapse operator for a position measurement, and the rate of spontaneous scattering gives information about the atomic position. To make these equations a bit cleaner, we can define a normalized collapse operator by
ΩR , (19.208)
19.5 Position Measurement via Excitation by a Local Probe Field where we have defined the integrated Rabi frequency ΩR := sZ \infty −\infty dz |Ω(z)|2. (19.209) Then in terms of the normalized collapse operators, the SME becomes
¯h[HCM, \rho]dt − Ω2R 2\Gamma H |A(z)|2
Z dΩJ A(z)e−ik\cdotr
dΩ . (19.210) and the Poisson means become
dΩ
Ω2R \Gamma
|A(z)|2
(19.211) The overall rate of information gain is thus given by the ratio Ω2R /\Gamma. Note that the two information-related terms have opposite effects: for a localized probe profile |A(z)|, the last (stochastic) term collapses (local- izes) the atomic wave function by multiplying by |A(z)| and renormalizing. By contrast, in the absence of photodetection events, the second term moves the atom away from the probe by transforming the proba- bility density according to \rhozz −\rightarrow \rhozz −2(Ω2R /2\Gamma)(|A(z)|2\rhozz + \rhozz|A(z)|2) dt, thus tending to reduce the population where |A(z)|2 is maximum.
and
amplitude in steady state is related to the ground-state amplitude by \psie(z) ∝Ω(z)\psig. The spontaneous- emission event then ‘‘flushes away’’ \psig(z), making \psie(z) the new atomic wave function. While the information is given in principle by the emission event, the information is ‘‘set up’’ by the absorption of the photon from the probe field. Correspondingly, the back-action on the quantum state due to the localized probe (i.e., the increase of the momentum width of the atomic state due to the position-space collapse) is due to absorption of a photon with a superposition of wave-vector orientations, as is consistent with having the localized probe field. Of course, as we saw from the quantum theory of imaged resonance fluorescence above, the emitted photon contains yet more center-of-mass information about the atom, beyond the fact that it has merely scattered an atom, as a consequence of the random direction of the photon recoil. In principle, you could ex- tract the most information about the atom by also imaging the resonance fluorescence, but if the fluorescence is merely detected without imaging or angle resolution, then we should trace over all possible photodetection angles in (19.210),
¯h[HCM, \rho]dt − Ω2R 2\Gamma H |A(z)|2
Z 1 −1
A(z)e−ikz cos \theta \rho dN, (19.212) so that
Ω2R \Gamma
|A(z)|2 dt, (19.213) and the last term in the SME puts the atom in an incoherent superposition of having recoiled in all possible directions, weighted by the correct probabilities. Note that we have also carried out the \phi part of the angular integral in the last term of (19.212), where
Z 2\pi
(19.214) is the effective angular distribution for the atomic resonance fluorescence, since the \phi angle is immaterial as far as the atomic dynamics are concerned.
19.5.3 Example: Gaussian Probe¶
Chapter 19. Position Measurement 19.5.2 Scanning Probe Field The fluorescent probe, as outlined above, only gives information about whether or not the atom is in the vicinity of the probe. To obtain a more standard position measurement, we can now consider the case of a moving probe field, where the center of the probe moves according to the trajectory zprobe(t). We will assume zprobe(t) to be a sawtooth function of constant velocity vprobe, but jumping discontinuously from zmax to −zmax at the end of each sweep. We will also assume the time for a single scan to be much slower than the time scale \Gamma−1 for the internal state to equilibrate, but we will assume it to be much faster than any motional time scale for the atom. We also assume that the atom will remain localized within the region (−zmax, zmax). The effect of the moving probe is to make the replacement A(z) −\rightarrow A[z −zprobe(t)] in the above SME. Performing a time average on the probe-raster time scale in the second term then amounts to replacing A(z−zprobe(t)) by a function that is approximately uniform over (−zmax, zmax) and zero elsewhere. Because of our assumption that the atom stays within the range of the probe, the second term has no effect on the atomic state, and can be dropped. What we are essentially saying is that the probe should always excite the atom equally no matter where it is, and thus there is no information to be gained by not detecting a photon. Thus,
Z 1 −1
A[z −zprobe(t)]e−ikz cos \theta \rho dN, (19.215) with
Ω2R \Gamma
|A[z −zprobe(t)]|2 dt. (19.216) Now we can see that zprobe(t) acts as an index for the displaced collapse operator A(z −zprobe(t)). If the probe raster time ∆tr is much shorter than both the time scale for atomic motion and the mean time between spontaneous-scattering events, but we carefully time-resolve the detection events, then as far as the motional dynamics are concerned, we can time-average the dynamics on time scales of ∆tr to write
Z 1 −1
A(z −zd)e−ikz cos \theta \rho dN, (19.217) where zd \in (−zmax, zmax), which is simply zprobe(t) evaluated at the time of the detection event, is a stochastic, random variable with probability density
|A(z −zd)|2 , (19.218) and the Poisson process now responds only to the time-averaged probe intensity,
Ω2R \Gamma Z \infty −\infty dz |A(z)|2 rect(z/2zmax)
(19.219) where the bracketed quantity represents the convolution of the probe-intensity profile |A(z)|2 with the time-
that the atom stays away from the edges of the scan range), and we have used rect(z) as the rectangular-pulse function of unit height and width. 19.5.3 Example: Gaussian Probe Again, to see how much information we are getting, we can compare the diffusion rate due to the measurement process to the rate of information gain. If we take the collapse operators A(z) to have a Gaussian profile, that is, we take the probe intensity to have the Gaussian form |A(z −zd)|2 ∝exp −2(z −zd)2 w 2 , (19.220)
19.6.1 General Remarks¶
19.6 Continuous Momentum Measurement by EIT where w0 is the beam-waist parameter for the Gaussian beam, we can then compare to the Gaussian form
0 ∆t. Noting that the average time ∆t between detection events is 1/\gamma (assuming unit detection efficiency of the radiated photons), the measurement strength in the formal white-noise limit is \kappa = \gamma/2w 2 0 . From our analysis of Eqs. (19.88), we conclude that the
However, this is only the diffusion rate due to absorption, which is where the position-dependent nature of the probe enters; the emission events cause additional diffusion. From Eq. (19.195), the diffusion rate due to spontaneous emission (i.e., the application of the e−ikz cos \theta factors), assuming an atomic dipole oriented along the z-axis, is
. (19.221) Thus, the effective measurement-information efficiency for this probe-measurement scheme is that fraction of the total diffusion rate that corresponds to the measurement gain (i.e., the absorption):
Dp Dp + DSE = 1 + 2w 2 0 k2/5. (19.222) In other words, the effective measurement efficiency goes down as the beam waist w0 becomes larger, because the information gained becomes smaller but the disturbance due to spontaneous emission is the same. Practically, w0 is limited to something on the order of \lambda, and for a very tight focus of w0 = \lambda, the effective measurement efficiency would be limited to a maximum of \etaeff = 6%, a rather low value. The efficiency is correspondingly further reduced by the detection efficiency. 19.6 Continuous Momentum Measurement by EIT As an alternative to continuous measurements of position, we can also consider schemes to continuously measure the momentum of a single atom. In some sense, continuous measurements of position and momentum are equivalent, since to some extent one can infer a momentum trajectory from a continuous record of position, and vice versa. In the simple but important example of the harmonic oscillator, position and momentum represent different yet equivalent directions in phase space. Electromagnetically induced transparency (EIT) provides a momentum-sensitive probe for a single atom that works without a cavity.16 19.6.1 General Remarks Recalling the phenomenon of EIT from Section 6.2.2, the susceptibility of an atomic gas of number density N for a weak probe field (field 2) due to the pump (field 1) is [from Eq. (6.70)]
ϵ0¯h
Ω2 , (19.223) where we have taken the limit of a strong pump field (large Ω1). We assume now that for an atom at rest,
counterpropagate, so that ∆1 = ∆+ k2v and ∆2 = ∆−k2v, where v is the atomic velocity in the direction of the EIT fields, and k2 is the wave number of the probe field. Then
ϵ0¯hΩ2
(19.224) and assuming that the ground-state relaxation rate \gammag is negligible, we see that the susceptibility is real and proportional to the atomic velocity. The atoms thus present a refractive index n = 1 + Re[\chi]/2 whose deviation from the vacuum value is proportional to the velocity. 16P. Rabl, V. Steixner and P. Zoller, ‘‘Quantum limited velocity readout and quantum feedback cooling of a trapped ion via electromagnetically induced transparency,’’ Physical Review A 72, 043823 (2005) (doi: 10.1103/PhysRevA.72.043823).
19.6.2 Homodyne Detection of the EIT Probe¶
Chapter 19. Position Measurement detector local oscillator EIT probe By measuring the phase of the probe beam using homodyne detection, we thus continuously extract momen- tum (velocity) information about the atom. Of course, the ‘‘number density’’ N for a single atom is small, but EIT functions to amplify the effect of the atomic momentum on the phase of the field, and so we need to treat this system more carefully below. 19.6.2 Homodyne Detection of the EIT Probe With direct detection of the EIT probe beam, the appropriate collapse operator for a detected photon in analogy with Section 18.2 is Cdirect = p
(19.225) where \Gamma2 is the partial decay rate on the EIT probe transition, \sigma2 := |g2\rangle \langle e| is the atomic annihilation operator for the EIT probe transition, and \alphaprobe is a complex number representing the coherent state of the probe field. Strictly speaking, this is the collapse operator for the mode of the probe field, and photons scattered into other modes must be treated by a separate detection process, as we will do later. This collapse operator already has the appropriate form for homodyne detection, as it represents the lack of knowledge about whether a detected photon came from the atom or from the probe field. We assume the pump field to be in a different mode, and we will otherwise ignore the pump except for its cooperative effects with the probe. When the probe field is then monitored via simple homodyne detection, as in the above diagram, the collapse operator is then modified to include the local-oscillator field as Chomodyne = p
(19.226) where we have already taken the limit as the reflection coefficient of the beam splitter vanishes, taking the local-oscillator amplitude to be correspondingly large (and absorbing the reflection coefficient into the local-oscillator amplitude \beta). Combining the two classical fields, we can write Chomodyne = p
(19.227) where \beta′ := \beta + \alphaprobe. This collapse operator has precisely the form of the collapse operator for simple homodyne detection, and thus the analysis for homodyne detection from Section 18.2 carries through here. Thus, from our previous analysis of homodyne detection, the SME becomes
p \eta\Gamma2H
\rho dW, (19.228) where H is the atomic Hamiltonian, including the interaction with the pump and probe fields, \phi is the phase of the combined field \beta′,
(19.229) and the efficiency \eta represents the fraction of photons radiated by the atom on the |g2\rangle −\rightarrow |e\rangle transition into the mode of the probe beam—recall that the phase shift of the probe beam is due to the interference of the dipole radiation and the original probe field. The corresponding scaled photocurrent (measurement record) is
D
2e−i\phiE dt + s \Gamma2 \eta dW, (19.230) so that we must still choose the local-oscillator phase \phi to obtain the appropriate information.
19.6.4 Spontaneous Scattering¶
19.6 Continuous Momentum Measurement by EIT 19.6.3 Adiabatic Approximation Since the atomic motion is much slower than the internal atomic dynamics, we can adiabatically eliminate the internal atomic state by replacing the internal atomic operators (namely \sigma2) by their steady-state values. The steady-state coherence from Eq. (6.68) on the probe transition |g2\rangle −\rightarrow |e\rangle is
(19.231)
assuming a counterpropagating pump-probe pair, so that ∆1 = ∆+ k2v and ∆2 = ∆−k2v, and keeping only the first-order velocity term,
Ω2 v = 4k2Ω2 maΩ2 p. (19.232)
p
s \Gamma2 \eta dW, (19.233) where the measurement strength is
2 \Gamma2Ω2 m 2a Ω4 , (19.234) where ma is the atomic mass. The SME (19.228) then becomes
p
(19.235) and the measurement record (19.233) can be rescaled to appear in a more standard form for a position-type measurement: dy := d˜r(t) \sqrt8\kappa\Gamma2
(19.236) This result is valid in the momentum range |p| ≪ maΩ2 8k2 p ∆2
(19.237) which follows if the second term of the denominator of Eq. (19.231) is to dominate the first. If we assume a small probe detuning ∆2 compared to \Gamma2, a moderate pump of Ω1 = \Gamma2 ∼20 \times 106 s−1, and we consider
19.6.4 Spontaneous Scattering The spontaneous scattering rate should be small, at least when the motion of the atom stays near p = 0, due to the nature of the dark state. To compute the rate of spontaneous scattering, we use the equation of motion
(19.238) as follows from the unconditioned master equation for EIT, Eq. (6.63). In steady state, this gives the excited-state population in terms of the coherence on the probe-transition:
2\Gamma2
(19.239)
19.6.5 Phase¶
Chapter 19. Position Measurement We can expand the steady-state coherence (19.231) to second order in momentum to obtain
maΩ2 p −32k 2
m 2a Ω2 p2 + O(p3), (19.240) and then put this into Eq. (19.239) to find
2 Ω2 m 2a Ω4 p2 + O(p3). (19.241) The steady-state scattering rate is then simply
2 (\Gamma1 + \Gamma2)Ω2 m 2a Ω4 p2 + O(p3), (19.242) so we see that the scattering rate vanishes to first order in the atomic momentum. However, to account for this effect, we should include an extra diffusion term to the SME (19.235)
p
Z 1 −1
(19.243)
extra spontaneous scattering is not detected. Note again that since the EIT measurement is dispersive, it scales as p, while the spontaneous emission scales as p2. Again, this is because the dipole-radiated field scales as p, and the phase shift is due to an interference with this field and the forward EIT probe beam, so that the phase shift goes as the product of the two fields, while the spontaneous emission goes as the square of the dipole-radiated field. 19.6.5 Phase It is illuminating to work out the relative phases of the atomic dipole, the EIT probe field, and the local- oscillator field. The phase of the atomic dipole, given by the phase of Eq. (19.232), is defined with respect to the phase of the EIT probe field. To lowest order in momentum, the atomic dipole (and thus the dipole field) is thus exactly in phase with the probe field. Further, we saw that the appropriate choice of local-oscillator phase was \phi = 0, so that the local oscillator is also exactly in phase with the atomic dipole and the driving field. This seems a bit strange: if the probe is, in fact, phase-shifted by the effective refractive index of the atom, then the atom should radiate in quadrature with the field, not in phase with it (radiation exactly in or exactly out of phase can affect only the amplitude, not the phase, of the probe field). Further, to detect the phase of the probe field, the local oscillator should again be in quadrature with the probe field, so that any phase shifts of the probe would act to modulate the detected intensity of the local oscillator. An easy way to resolve this difficulty is to note that to have a strong coupling to the atom, the EIT probe field should be tightly focused onto the atom. If we take as a concrete example a Gaussian beam, we may write the probe field as
−ˆx + x z −iz0 ˆz w0 w(z) exp − r2 w2(z) exp ikz −i tan−1 z z0 exp ik r2 2R(z) , (19.244) where the beam propagates along the z-direction and is (mostly) polarized along the −x-direction, w0 is the Gaussian beam-waist parameter that characterizes the beam waist at the focus, z0 = \piw 2 0 /\lambda is the Rayleigh
p 1 + (z/z0)2 is the z-dependent spot size, and
solves the electromagnetic wave equation in the paraxial approximation (i.e., as long as the divergence angle of the beam is not too large). The important thing to notice is the second exponential factor, which gives the longitudinal phase. There is the usual plane-wave-type phase of ikz, but there is also the Gouy phase
Gouy phase is generic for focused beams, and plays an important role here.
19.6.6 Detection Efficiency¶
19.6 Continuous Momentum Measurement by EIT The dipole field lacks this Gouy phase, with a phase varying as ikz along the z-axis. Similarly, we take the local oscillator to be collimated (but still Gaussian), so its Gouy phase varies negligibly over the scale of the optical apparatus. Thus, even though the dipole, EIT-probe, and local-oscillator fields are all in phase
field, it has accumulated an extra phase shift of −i\pi/2. Thus, as we expect, the EIT probe is in quadrature with both the atomic-dipole and local-oscillator fields. From this, we can view the homodyne detection slightly differently. Rather than regarding the detection as a measurement of the phase of \sigma2, we can regard it as a measurement of the phase of the phase-shifted
analysis above showed the phase of \sigma2 to be near zero, and the same choice of phase was best for the local oscillator, while the phase of −i in the EIT probe here is due to the Gouy phase accumulated as the probe beam travels from the atom to the detector. Viewed thusly, since the homodyne-detection signal measured D
E in the analysis above, here it measures
= D
E , and thus the conclusions above still obtain: because the probe and local-oscillator fields are in quadrature, the homodyne measurement rejects any contribution to the measurement signal from the probe. Similarly, viewed this way, the analysis for balanced homodyne detection as in Section 18.2.6 carries through here, if we view the balanced homodyne detection as a measurement of the probe phase rather than the atomic phase. Then the same SME and measurement record result, except that there is no need to subtract a dc offset from the measurement signal. 19.6.6 Detection Efficiency The detection efficiency \eta here is simply the probability that a photon radiated by the atom is scattered into the mode of the EIT probe. As an example case, we will take the EIT probe again to be a Gaussian beam, and compute explicitly the overlap between the probe and dipole waves. It is sufficient to consider the overlap in the far field, for which the Gaussian beam (19.244) becomes
z0 z exp −z 2 0 r2 w 2 0 z2 , (19.245) noting that we have dropped the polarization and phase factors, as they will be irrelevant for the mode overlap (within the paraxial approximation, they will exactly match the same factors for the dipole wave). We can then write this field as a normalized field mode function p
2z0 \sqrt 2\pi w0z exp −z 2 0 r2 w 2 0 z2 , (19.246) normalized so that at any fixed z, Z dx Z
(19.247) Assuming a linearly polarized atomic dipole, the usual field dipole pattern is p
r
s 8\pi y2 + z2
(19.248) but where \theta and \phi are defined with respect to the polarization (x) axis, not the z-axis. In the paraxial approximation, we may write this field in terms of a normalized spatial distribution (by dividing by z) as p
z r 8\pi 1 −x2 2z2 . (19.249) Then the efficiency is the overlap integral of the Gaussian field with the dipole field:
Z dx Z dy p
\sqrt 3w0(4z 2 0 −w 2 0 ) 8z 3 \approx \sqrt 3 \lambda 2\piw0 . (19.250)
Chapter 19. Position Measurement
good), the efficiency is 27%.
19.7 Exercises 19.7 Exercises Problem 19.1 Show that the Weyl correspondence (Section 4.3.5) for the stochastic master equation for position measurement,
\sqrt
\sqrt 2\kappa
dW(t) (19.251) (without Hamiltonian evolution) gives the Fokker-Planck equation for the Wigner function with no drift and a momentum diffusion coefficient of D = 2¯h2\kappa, plus a stochastic driving term:
p W(x, p) dt + \sqrt 8\kappa
W(x, p) dW(t). (19.252) Problem 19.2 Show that the Weyl correspondence (Section 4.3.5) for the stochastic master equation for position- squared measurement,
\sqrt
\sqrt 2\kappa
x2 \rho dW(t) (19.253) (without Hamiltonian evolution) gives the Fokker-Planck equation for the Wigner function with no drift and a position-dependent momentum diffusion coefficient of D = 8¯h2\kappax2, plus a stochastic driving term:
p W(x, p) dt + \sqrt 8\kappa
x2 −
x2 −¯h2\partial 2 p ! W(x, p) dW(t). (19.254) Problem 19.3 Consider a particle subjected to a noisy potential of the form
dt . (19.255) (a) Why is it appropriate to write the potential in Stratonovich form, rather than It¯o form? (b) Write down a stochastic Schrödinger equation for the particle, convert it to It¯o form, and then use the result to derive a stochastic master equation. (c) Show that in the ensemble average, this SME is equivalent to the unconditioned SME for position measuremeent. (In the conditioned case, the two master equations are not equivalent; for example, the noisy potential still generates a linear SME.)