## An introduction to quantum measurement theory - UQ eSpace

Chapter 1. An introduction to quantum measurement theory. In this introductory chapter we present those aspects of quantum measurement theory that we use in .

Chapter 1

An introduction to quantum measurement theory

In this introductory chapter we present those aspects of quantum measurement theory that we use in this thesis to discuss the quantum Zeno effect, both theoretically and to make experimental proposals for its testing.

1.1

Measurement in quantum mechanics

That the Schr¨ odinger equation  governs the dynamics of all closed physical systems is one of the fundamental postulates of quantum mechanics. In the Schr¨ odinger representation this equation is i¯h

∂ ˆ |ψi = H|ψi, ∂t

(1.1)

ˆ is the (presumed) time independent Hamiltonian of the system of interest described by the where H state vector |ψi. (We discuss our operational definition of these terms below.) Under this equation the evolution of the system is fully unitary and time reversible. The state at time t is related to the the state at time t = 0 by the evolution operator U such that |ψ(t)i

=

U |ψ(0)i

=

e− h¯ Ht |ψ(0)i. i

ˆ

(1.2)

This evolution is linear, and therefore the state of the system can always be described by a linear combination of the members of any particular eigenbasis of interest. The adoption of the Schr¨ odinger equation as a postulate necessarily entails the acceptance of the existence of linear superpositions of eigenstates for arbitrary closed systems. This is of considerable interest as the postulated Schr¨ odinger equation contains no parameters concerning the size or complexity of the system of interest, and thereby 1

CHAPTER 1. AN INTRODUCTION TO QUANTUM MEASUREMENT THEORY

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claims applicability to all closed systems. In the Chap. (9) we intimate that the Schr¨ odinger equation could be applied to the Universe as a whole, with the conclusion that the Universe exists in a coherent superposition of states. This prediction is difficult to reconcile with observations of our classical Universe.

There are two major classes of systems which are generally observed not to be found in linear combinations of eigenstates. The first of these are macroscopic systems such as physicists or dust motes and other inconsequential objects. Such macroscopic systems consist of many microscopic systems interacting in such a way as to smear out their mutual quantum coherences and thus to render unobservable the superposition of their eigenstates. The extreme complexity of these interactions makes using quantum mechanics prohibitively difficult. Because of this, physicists make further postulates regarding the nature of macroscopic objects, and these postulates constitute classical mechanics. (Of course, this description reverses the usual order of postulation.)

The reconciling of these two paradigms is itself a difficult and active area of research. Classical mechanics forbids the existence of macroscopic superposition states, and in effect sets all quantum coherences to be identically zero. On the other hand, quantum mechanics (based on the Schr¨ odinger equation) ensures that the quantum coherences are faithfully maintained even though smeared out and thus predicts that systems of arbitrary size will display behaviour typical of quantum systems (such as recurrences for example). A fruitful approach to resolving this contradiction between these two fields consists in treating the interaction between a microscopic system, governed by quantum mechanics, and a macroscopic system referred to as the environment described semi-classically. Studying such a combination of micro- and macro-systems assists us in understanding how classical mechanics arises in a quantum universe. (See for example [28, 10].) Investigations such as these will be further considered in this thesis.

The second area in which the Schr¨ odinger equation is observed to be inapplicable is a subset of the system-environment interaction considered above. In this case, we specify the dynamics of the environment in such a way that part of it constitutes a semi-classical meter, or detector. We further carefully design the system-meter interaction so as to provide a channel along which information can flow and be amplified from the microscopic system so as to be reflected in the state of the macroscopic meter. After the (as yet unspecified) interaction and amplification has been completed we say that a measurement has been performed. Such a measurement interaction is observed to be non-unitary, and thus cannot be described by the Schr¨ odinger equation. An accurate measurement results in a reduction of the linear superposition of eigenstates in an arbitrary eigenbasis to a single eigenstate in a “pointer” basis  specified by the measurement interaction. The modelling of this measurement interaction constitutes a large part of this thesis.

1.1. MEASUREMENT IN QUANTUM MECHANICS

1.1.1

3

The Heisenberg “cut” and related measurement issues

We note that while we do model various measurement interactions in this thesis, we emphatically do not address the question of what constitutes a measurement, or when a particular interaction can be considered to be a measurement. This question fully encapsulates the issues raised above about the placement of the divide between quantum and classical mechanics. This divide is referred to as the “Heisenberg cut”. The Heisenberg cut  is used to divide the realms of the microscopic world in which quantum mechanics applies and the realm in which classical mechanics is presumed to apply. As intimated above, the location and modelling of this divide is an active area of research. A consideration of the act of performing a measurement on a quantum system brings the issues into sharp focus. An experimenter seeking to perform such a measurement has no direct access to the quantum system, but only to the measuring device being used. (This applies even when we attempt to observe photons directly for instance. In this case the observing device is the macroscopic lens and retina of our eye.) The observer builds a careful channel along which information can flow from the quantum system, through various amplification stages until it impinges upon the observer’s awareness. The observer is aware that every stage of the channel can be modelled quantum mechanically, including the observer. And ¨ thus, according to Schrdinger’s equation, at no stage is a nonunitary interaction that might constitute a measurement ever effected. Every stage in the measurement chain must then be described by a superposition of eigenstates. However, most observers would agree that they themselves do not exist in a quantum superposition of states. This is established by introspection, or by an observation of oneself. For all practical purposes, we generally treat most of the stages of the measurement chain as being well described by classical mechanics, and thus, not existing in some superposition of states. This then faces us squarely with the conundrums neatly summed up in the “Schr¨ odinger’s cat”  and “Wigner’s friend” paradoxes. (Attempts to resolve these paradoxes using metaphysics are, we believe, flawed and based on an alleged poor present understanding of the tools of quantum mechanics. We discuss this assertion in the epilogue and conclude that more attention must be given to developing better tools of quantum mechanics and a better understanding for these tools before any attempt at building a quantum metaphysics will be successful.) For the purposes of this thesis, we are content to assert operational definitions as required. In view of this we make the empirical observation that we are best described classically, and thus place the “Heisenberg cut” below us in the measurement chain. Where it is placed is relatively arbitrary. If it is placed at the level of the system itself, then we effectively model a measurement interaction as an instantaneous collapse of the system state. This instantaneous nonunitary collapse of the system state significantly disturbs the evolution of the system of interest and can’t be described by the Schr¨ odinger equation. In effect, we have two mechanisms by which a quantum state can evolve - the first described by the Schr¨odinger equation, and the second described by this nonunitary measurement interaction. (This

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disturbance, or measurement back action, when applied directly to the system of interest generates the elementary quantum Zeno effect as detailed below.) However, we generally find it more useful to model the quantum system as weakly interacting with another, larger system, the detector, and treat this larger system as being the first part of the measurement chain. This detector can be modelled pseudo-classically, and by this we mean that it possesses certain classical attributes. We choose to apply the “cut” to the detector, whose semi-classical attributes [discussed further in Chap. (3)] are chosen to minimise (but not ignore) the measurement backaction onto the system. In what follows we develop an operational definition of measurement, and along the way, define terms like state of a system, ensemble, measurement device, and so on. This operational definition is sufficient for us to build an intuitive understanding of the theory and enable us to meaningfully compare our predictions with experiment. To a large part we follow the approach referred to as the Copenhagen interpretation of quantum mechanics, though without delving (in this thesis) to deeply into the “meaning” of the terms employed in quantum theory. (Further discussion of some aspects of quantum metaphysics takes place in the epilogue.)

1.2

A guide to quantum measurement theory

We introduce quantum measurement theory by means of the states, effects and operations approach. For this treatment we follow the treatment of Kraus , but make use of the reformulation by Milburn [51, 17] of this work using Dirac notation. This methodology is useful for later generalization, and serves to introduce many of the techniques of quantum measurement theory. We note that the techniques presented in this section are used to model measurement outcomes. This measurement approach does not make any pretence at modelling the interaction between a system and a detector, and effectively, only models the outcomes of measurements performed on a quantum system. We turn to formulating models of a measurement interaction in Chap. (3).

1.2.1

Instantaneous, accurate measurements - elementary measurement theory

In this section we introduce the elementary quantum measurement theory of instantaneous, arbitrarily accurate measurements. We consider an ensemble of many quantum systems (possibly infinitely many) that have been subject to a well determined “state preparation procedure” specified by the construction of some macroscopic physical device. We then ascribe a “state” to the members of the ensemble. We further assume that this state can be described by density operators in a Hilbert space in the usual manner. Kraus then introduces the empirical fact that macroscopic measuring instruments exist and are designed so that they internally register an “effect” after an interaction with an individual quantum system. It is empirically

1.2. A GUIDE TO QUANTUM MEASUREMENT THEORY

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observed that, after such an interaction the effect apparatuses are “triggered with reproducible relative frequencies”  by the microsystems prepared in the above ensemble. We further assume that we can model these effects through the use of projection operators on the Hilbert space possessed by the system, and that the above relative frequencies of empirical observation can be determined by tracing the projection operator with the ensemble density operator over the Hilbert space. The above assumptions are best clarified by using a specific example. Consider the arbitrarily accurate measurement of a physical quantity represented by the operator Aˆ with eigenstates |ai on an ensemble of systems described by the density operator ρˆ, ρˆ =

X

ha|ˆ ρ|a0 i|aiha0 |.

(1.3)

a,a0

We do not detail the measurement interaction, and only specify the effect to be, for example, a projection |aiha|. After the completion of the undescribed measurement interaction, we postulate the measured value of the physical quantity Aˆ to be a (one of the eigenvalues) with probability P (a) = tr(ˆ ρ|aiha|).

(1.4)

The application of the projection postulate is here assumed to be instantaneous. The interaction with the measurement device does not leave the state of the system unaffected. There is always a measurement backaction onto the state of the system. Consider the ensemble of systems introduced above after the interaction with the effect device. We are perfectly justified in treating this additional interaction with the effect device as just another stage in the state preparation of the system, and, in view of the definition introduced above, we expect the system to be described by a new ensemble after the measurement interaction. We are naturally interested in the form of this new ensemble, but to describe it we must distinguish two experimental situations. Consider the case where the experimenter has perfect information about the outcome of the measurement interaction, then a “selective operation” is said to have occurred. This selective operation is also defined operationally in the following manner. After the measurement interaction, the experimenter selects those members of the ensemble that gave a particular result, and discards the rest. In this case, the outcome is a new ensemble whose members have all been through a special state selection procedure depending on the outcome of the previous measurement. In this case, we say the ensemble of systems has been partitioned according to the results of the measurement. The new ensemble corresponding to the result a is related to the old ensemble by the “selective operation” ρˆ(s)

=

= =

φ˜I ρˆ ³a ´ tr φ˜Ia ρˆ |aiha|ˆ ρ|aiha| tr(ˆ ρ|aiha|) |aiha|,

(1.5)

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where we introduce the operation φ˜Ia depending on the result a. The operation φ˜Ia is referred to as a super-operator or sometimes as a Liouville-type operator. Here we make explicit use of the first representation theorem of Kraus  which relates the operation to the effect. This theorem also specifies the conditions that must be satisfied by each of these constructions, and shows that any operation on a density operator can be written in terms of ordinary operators on Hilbert space. The partitioning of the ensemble is commonly referred to as the “collapse” of the wavefunction and simply constitutes a replacement of one wavefunction by another as a result of the additional information we have about the measurement outcome. The above measurement model based on the projection postulate  forms the simplest model for measurement outcomes. Upon the completion of the above measurement interaction where full account is taken of the readout value, any subsequent measurement performed immediately after the first will, if accurate, obtain the same eigenvalue −(a), as a result. In this case, the measurement interaction is said to be a Type I measurement  as it does not destroy the state of the system just measured. In such a case we see that the post-measurement state is left in the eigenstate |ai even for perfect projective measurements. This is not always the case. For instance, photon detection destroys photons upon detection. A measurement interaction that destroys the state of the system is referred to as a Type II measurement interaction. We will have need to distinguish between these measurement interactions in this thesis. A Type II measurement might destroy an arbitrary state |ai to leave the state |a0 i, and so, in this example, is described by the post-measurement ensemble ρˆ(s)

=

φ˜II ρˆ ³a ´ tr φ˜II ρˆ a

0

= =

|a iha|ˆ ρ|aiha0 | tr(ˆ ρ|aiha|) 0 |a iha0 |,

(1.6)

where again we employ the first representation theorem. In Type II interactions, subsequent measurements of the observable Aˆ will not reproduce the previous read out. However, a “collapse” of the wavefunction is still said to have occurred. The above description is of a selective operation where the information about the microscopic system is channelled to the macroscopic realm. The experimenter has information about the measurement outcome - (a). If on the other hand we take no account of the experimental result, and discard all information about the outcome of the experiment then we must describe a measurement interaction performing a “non-selective” operation. A channel still exists by which this information might be obtained, but we do not avail ourselves of this opportunity. Instead we choose to dissipate the information into the environment. Depending on the environment and the experimental set up being considered, this dissipation of the information might, or might not be irreversible. For example, we might dissipate photons into the environment, and thus perform a measurement interaction describing a nonselective operation, but there is nothing (in principle) preventing us from deploying a photo-detector several light

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years away to finally complete a delayed measurement interaction and thus complete a selective operation. In the case of an interaction describing a nonselective operation, the ensemble is not partitioned, and the new state of the ensemble contains all the members of the old ensemble. Under the new expanded state preparation procedure the state of the ensemble is given by (for Type I measurements) ρˆ(N S)

Φ(N S) ρˆ X = φ˜Ia ρˆ

=

a

=

X

|aiha|ˆ ρ|aiha|

a

=

X

P (a)|aiha|,

(1.7)

a

where setting Φ(N S) ρˆ =

P ˜I ˆ employs the second representation theorem due to Kraus . The a φa ρ

corresponding result for Type II measurements is ρˆ(N S) = |a0 iha0 |

(1.8)

reflecting the fact that while we have no information about the outcome of the measurement, we do know that, for the example considered above, the measurement interaction destroyed the initial state and left the system in state |a0 i. The above postulates involve the use of projection operators to model effects (here for example |aiha|) and the corresponding “collapse” of the wavefunction, and are sufficient to model the outcomes of single, instantaneous, accurate measurement interactions. Predictions based on these postulates are very well confirmed by experiment. However, there are some immediately apparent drawbacks with the use of these postulates to describe measurements in quantum mechanics. The first of the drawbacks to the above approach is that it does not model the actual measurement interaction between the system and detector. We address the issue of modelling specific system-detector interactions in Chap. (3). A worse drawback is that this approach gives no criteria for distinguishing whether a particular interaction between two arbitrary systems constitutes a measurement interaction or not. When does an interaction between a system and an apparatus constitute a measurement, and when does it simply result in unitary evolution. We answer this question operationally and apply what is referred to as the “Heisenberg cut” in order to resolve the issue of just when our measurement device does enact a measurement and just when it interacts unitarily with the system of interest. The application of this procedure has been discussed in the preceding section.

1.2.2

Inaccurate measurement interactions

A further drawback in the above approach is that the simple projection postulate |aiha| cannot well describe either inaccurate measurement interactions or continuous measurement interactions. For example, consider position measurements of a free particle. The use of the projector |xihx| to model accurate

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position measurement outcomes gives the correct distribution for x ˆ but predicts that the particle has acquired infinite energy from the instantaneous interaction with the measurement device. This is a result of the uncertainty relations between the observables x ˆ and pˆ which establish that a zero uncertainty in position must reflect an infinite uncertainty in momentum. To describe such continuous measurements we must also have mechanisms for describing inaccurate measurement interactions. We now consider such mechanisms. Here we briefly generalize the simple projective measurement approach considered above to treat measurements of finite accuracy. (Again we follow the treatment of Refs. [11, 51, 17] in using the terminology of effects and operations.) To achieve our aims Eq. (1.4) must be generalized, and the approach we take is to explicitly recognise that typical measurement devices make inaccurate measurements. To model this, we parameterize the accuracy of the measurement effect device by some parameter σ. The outcome of the measurement is still taken to be some result a. The probability of obtaining the measurement result a is Pσ (a)

³ ´ tr ρˆFˆσ (a) ¢ ¡ ρΥ†σ (a) . tr Υσ (a)ˆ

= =

(1.9)

Here, Fˆσ (a) = Υ†σ (a)Υσ (a) is a positive, bounded, self-adjoint operator on Hilbert space and is referred to as an effect. This is not the most general formulation of a measurement that could be made. This can be seen by noting that we have specified the effect Fˆσ (a) giving the outcome a. In reality, there might be many different operators Υσ (a) that generate the same effect Fˆσ (a) and give the outcome a. We later discuss the generalizations required to take our ignorance of the specific operators used in the measurement interaction into account. The effect defined above can easily model both non-destructive [Type I] measurements and destructive [Type II] measurements. As an example of this we consider photon detection. This example will arise later in the thesis. To model a quantum nondemolition or non-destructive counting of photons (a Type I measurement) we choose ˆ σ (m) = Γ

¸1/2 ∞ ·µ ¶ X k σ m (1 − σ)k−m |kib hk|. m

(1.10)

k=m

ˆ → |mihm| which gives the usual non-destructive In this equation it is evident that for σ → 1, Γ projection operator. This equation has an obvious interpretation when we interpret σ, the accuracy of the measurement, as the probability of detecting a photon, and conversely, (1 − σ) is the probability of not detecting a photon. For the case of destructive photon counting (a Type II measurement) we have ˆ σ (m) = Γ

¸1/2 ∞ ·µ ¶ X k σ m (1 − σ)k−m |k − mib hk|. m

k=m

(1.11)

1.2. A GUIDE TO QUANTUM MEASUREMENT THEORY

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ˆ → |0ihm|, the expected projection operator. In both cases the related Again for σ → 1, we see that Γ effect is Fˆσ (m)

= =

ˆ †σ (m)Γ ˆ σ (m) Γ µ ¶ ∞ X k σ m (1 − σ)k−m |kib hk|. m

(1.12)

k=m

Here, it is evident that the effect Fˆσ (m) does not depend whether the photon number measurement was destructive or not. If we were ignorant of which measurement interaction was being employed we would need to generalise our approach. This is discussed below. We further note in the above the general result I=

X

Υ†σ (a)Υσ (a).

(1.13)

a

The First Representation theorem of Kraus  also shows that it is not only projection operators that can describe effects. As defined in the previous section, an effect can always be related to a super-operator φ˜σ (a) (referred to as an operation) which operates on the system density matrix to generate the post-measurement state ρˆ(S) σ =

ρ φ˜σ (a)ˆ , tr(φ˜σ (a)ˆ ρ)

(1.14)

where ρ = Υσ (a)ˆ ρΥ†σ (a) φ˜σ (a)ˆ

(1.15)

Here the ensemble has been partitioned on the results of the measurement - that is the operation φ˜σ (a) represents a measurement interaction treated selectively. We further note that the effect Fˆσ (a) does not uniquely determine the operation φ˜σ (a), and this reflects the fact that “different effect apparatuses . . . need not perform the same operation” . The operation and the effect are related to each other and the probability Pσ (a) of achieving the result a as ³ ´ ³ ´ Pσ (a) = tr φ˜σ (a)ˆ ρ = tr ρˆFˆσ (a) ,

(1.16)

as can be seen from Eqs. (1.9) and (1.15). As noted, this relation is not one to one as there may be many operations for a given effect. What this means in real terms is that differing measurement devices may yield the same measurement statistics for a. This is further discussed below. Should a non-selective inaccurate operation have been performed, the post-measurement state generalizing Eq. (1.7) is ρˆ(N S) =

X

ρ = φ˜σ ρˆ, φ˜σ (a)ˆ

a

P where φ˜σ = a φ˜σ (a) is the nonselective operation for the measurement.

(1.17)

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We have previously noted that the effect Fˆσ (a) does not uniquely determine the operation φ˜σ (a), and have given examples of this above. If we are ignorant of the specific operation used in a measurement interaction, we must generalize our approach to consider a two parameter operation  ρ = Υσ (a, α)ˆ ρΥ†σ (a, α), φ˜σ (a, α)ˆ

(1.18)

where α labels the various operations that can be used to generate an outcome a. In this case the effect is given by Fˆσ (a) =

X

Υ†σ (a, α)Υσ (a, α).

(1.19)

α

The probability of getting the outcome a given ignorance of the specific measurement interaction is then as given above. The above is adequate to get us started on the measurement problem and we introduce further details as needed.

1.3

Quantum trajectories

It has been recently asserted  that one of the interpretational failings of the Copenhagen interpretation is in the application of quantum mechanics to individual systems. In this thesis we will be modelling the evolution of individual systems a number of times, both when such systems are evolving freely, and when subject to either measurement or damping interactions. We briefly introduce the techniques for doing this now. We use the technique of quantum trajectories [29, 30, 31, 40, 36, 35] as another way of viewing nonunitary quantum dynamics which lies closer to our intuitive understanding of the experimental context. The operational definition of state vectors and density matrices introduced above, employed the concept of a “state preparation” procedure. This procedure created an ensemble of systems each possessing a certain state. This assumption could always be tested through the use of effect apparatuses to evaluate reproducible measurement outcomes. Recently developed techniques in quantum mechanics make it routine to perform measurements on well defined single quantum systems. The question then arises as to how we apply the above formulation to follow the evolution of a single system subject to some evolution process. This process could be either the usual unitary evolution, a measurement interaction, or a general nonunitary evolution. The method employed is similar to that introduced above. We take a single quantum system that has been through some state preparation step. The state of the system can then be interpreted as above as representing a collection of systems, all of which have been through the same procedure. We then wish to follow the evolution of the system, and note that this evolution might be any, or all, of the three types described above. However, each is treated in an equivalent way, and is represented by a further state preparation step. This evolution step takes one

1.3. QUANTUM TRAJECTORIES

11

density matrix to another, and as defined above, is represented by an operation. In other words, to evolve the system forward in time we apply an operation representing any of the three possibilities introduced above as desired. By this means we can evolve the quantum system forward in time under any sort of desired evolution. The new feature of the quantum trajectories approach is to allow the path of the evolution of the system to be determined at each infinitesimal time step, by the properties of the system at that time step. It is this feature that permits us deep insight into the evolution of quantum systems. It is routine to evolve a system forward in time through the use of an operation, and this has been done ever since the evolution operator was introduced as part of the solution to the Schr¨ odinger equation (1.2). However, if we put in these operations by hand, then we constrain the evolution of the system to fit with our conceptions, rather than allowing the evolution of the system to frame our conceptions. A better way is to define the initial state of the system, and as many operations as are required to specify all the possible evolutions that the system might undergo. We also need a scheme to evaluate at any time what the system is likely to do. We combine these elements, in forming a quantum trajectory for the evolution of the system. The trajectory is broken into (generally) infinitesimal time steps, and the probability of each of the operations being performed is evaluated based on the state of the system at that time. We can then employ a random number generator to evaluate which of the relevant operations is to be enacted at any particular time. This decision can be determined solely by the state of the system (if so desired). We apply the chosen operation, and repeat the procedure. This is the essence of the quantum trajectories approach. The form of the trajectories can be counterintuitive to physicists trained “from birth” in using nonselective master equation methods, but it is precisely this feature that makes this method so valuable in generating insight. Indeed, H.Carmichael has intimated that he foresees a time when quantum mechanics is initially taught using the quantum trajectories approach, and is only later generalized to consider the usual master equations. The above has been a brief summary of the philosophy underlying the technique of quantum trajectories. In what follows we go into detail sufficient for the needs of this thesis. We follow the formalism developed by Srinivas and Davies [32, 33, 34, 38] who (I believe) developed the formal solution of the master equation shown below. Our main motivation comes from contact with Carmichael who freely made available his lecture notes on this topic [30, 29] and who focussed on the quantum dynamics of the system, as opposed to the view emphasised by Srinivas and Davies which treated the interaction between the radiation field and a photo-detector (for instance). A useful guide to the use of these techniques to model photon detection is given in Milburn and Walls . A similar approach is also nicely covered in later work by Ueda . The literature in this field is extensive due to wide applicability of the insights offered. Recent work includes Refs. [31, 40, 36, 35]. We will develop this approach in various forms in this thesis as required, and introduce further references in later sections. Following Carmichael’s emphasis on “unravelling” master equations, we consider a generic master

CHAPTER 1. AN INTRODUCTION TO QUANTUM MEASUREMENT THEORY

12

equation of the form ρˆ˙ (t) = (L + J )ˆ ρ(t),

(1.20)

where L and J are super-operators equivalent to the operations introduced above. These operators can be chosen to represent any one of either free system evolution, the evolution of a damped system (in the Markoff approximation), or of measurement effected system dynamics. As we are interested in the dynamics of the system, we do not explicitly model the detector, or the environment under the Markoff approximation. An example that will come up in this thesis is the detection of photons emitted from a cavity mode of the electromagnetic field. In this case we would treat the environment as being configured as a detector, and every photon emitted from the system is collected and measured (counted) by the photodetector. (As usual the experimenter has the choice of either discarding the results of the readout or not.) To model such photon detection we would set Lˆ ρ(t) R J ρˆ(t)

R R i ˆ ρˆ] − ρˆ − ρˆ = − [H, h ¯ 2 2 = γa† a = γaˆ ρa† ,

(1.21)

where γ is a damping rate of the cavity into the field. In this equation the value of R is chosen to satisfy tr(Rρˆ) = tr(J ρˆ).

(1.22)

It is straight forward to show by substitution that these assignments generate a master equation of the usual form for a damped cavity, namely ¢ γ¡ i ˆ ρˆ] + 2aˆ ρa† − a† aˆ ρ − ρˆa† a . ρˆ˙ = − [H, ¯h 2

(1.23)

As is usual in this approach, the operation J models the detection of a photon coming from the cavity. If we detect a photon then we can use the techniques of the previous section to calculate the new state of the cavity field. The detection of a photon represents a new state preparation step acting on the state of the cavity, and this preparation step is described by the operation J . This is seen through the effect of J on the number state  J |nihn| = nγ|n − 1ihn − 1|.

(1.24)

To interpret the operation L it is useful to consider the new operation it ˆ Rt it ˆ Rt eLt ρˆ = e− h¯ H− 2 ρˆe h¯ H− 2 .

(1.25)

It is then possible to show that this operation does not change the state of the system, and is proportional to the probability of not detecting any photons in a time t. This can be seen through the application of this state to a number state  giving eLt |nihn| = e−γnt |nihn|.

(1.26)

1.3. QUANTUM TRAJECTORIES

13

We are now equipped to write the formal solution to the master equation (1.20) as ρˆ(t)

= e(L+J )t ρˆ(0) Z ∞ Z t X dtm = m=0

0

0

tm

Z dtm−1 · · · 0

t2

dtm1 × eL(t−tm ) J · · · J eL(t2 −t1 ) J eLt1 ρˆ0 .

(1.27)

We note that with the above interpretations of these operations we can interpret this expansion as a sum over many possible histories of photon emission events from the cavity. Eq. (1.27) is the general solution to the non-selective master equation and thus describes the case where the experimenter has discarded all information deriving from the detector. This discarding of information can be seen in this equation as the sum over m, all possible detection events, and the multiple integrals over all possible detection times tm . An experimenter is always able to keep this information, if so desired, and if this information is retained for a particular system, then we can specify the selective state of the system at time t, given the emission of m photons at times t1 ≤ t2 ≤ · · · ≤ tm ≤ t, as ρˆ(m, t, t1 , · · · , tm ) P (m, t, t1 , · · · , tm )

= P −1 (m, t, t1 , · · · , tm ) × eL(t−tm ) J · · · J eL(t2 −t1 ) J eLt1 ρˆ0 i h = tr eL(t−tm ) J eL(tm −tm−1 ) J · · · J eL(t2 −t1 ) J eLt1 ρˆ0 .

(1.28)

This expansion can be understood in terms of the state preparation procedures introduced in the previous sections. We are observing a single system, and have perfect information about all detection events originating in that system. This history allows us to apply the required state preparing operations to exactly model the state of the system at any time. By doing this we describe the evolution of a single system, the state of which is conditioned on the entire history of measurement results. If all the information lost from the system is recorded it is possible to represent this path as a stochastic evolution of pure states and this has significant calculational advantages as discussed below. We can then model the behaviour of an ensemble of systems from the sum of T independently generated trajectories. The sum over trajectories is equivalent to discarding all information about the history of measurement results. We illustrate this later in this thesis when we compare such a sum over trajectories to the non-selective evolution calculated from the usual master equation. The algorithms for doing this have been sketched in broad strokes above, and involve a calculation at each infinitesimal time step of the likely next event in the evolution of the system. We allow the state of the system at any time to specify the probabilities that a photon will be emitted, and employ a random number generator to decide this issue. This process calculates a stochastic conditioned trajectory for the individual system. Each history is itself a random event and we can then talk about stochastic paths of states. This equation describes the trajectory of the system at any time given the observations of m, t1 , · · · , tm . In generating a particular quantum trajectory we do the reverse. We generate the m, t1 , · · · , tm trajectory with probability P (m, t, t1 , · · · , tm ). We take the minimum resolution time of the photon detection apparatus to be τ . In this time, the apparatus can distinguish between the detection

CHAPTER 1. AN INTRODUCTION TO QUANTUM MEASUREMENT THEORY

14

of m = 0 photons or m > 0 photons, but can’t distinguish between m = 1 and m = 2 photons, for example. The probability for detection of m = 0 or m > 0 photons is P (m = 0, τ )

=

P (0, τ )

P (m > 0, τ )

=

1 − P (0, τ ),

(1.29)

where P (0, τ ) is calculated from Eq. (1.28). After any resolution time τ , we decide whether the system has emitted a photon or not, with the probabilities given above. There are a number of advantages to this approach. We gain special insight into quantum systems when we follow the selective evolution of a single system as opposed to the averaged, non-selective evolution of an ensemble. There can be a computational advantage for systems with large numbers of levels . In our case, the action of both L and J on ρˆ is to take pure states to pure states so we can model the state of the system as a vector, rather than having to use the density matrix. The computation of a trajectory is simplified further by noting that often, the detection of a photon collapses the system to a known initial state, and every time this happens, the subsequent evolution away from the initial state follows an identical path. Every trajectory can be easily generated from the trajectory that has no emission of photons. [We give an explicit example of this in Chap. (5)] Also we can recover the non-selective evolution trajectory by summing over a large number of trajectories. Finally we note that the non-selective evolution can be obtained directly by treating the non-selective density matrix as a special case of a mixed state and iterating its trajectory as a mixed state density matrix. To generate the non-selective evolution we simply discard all information about the emission or non-emission of photons at every step. This requires the addition of the no-emission density matrix and the one-photon-emission density matrix with corresponding probabilities at every step.