Quantum continuous measurements: The stochastic Schrödinger equations and the spectrum of the output
The stochastic Schrödinger equation, of classical or quantum type, allows to describe open quantum systems under measurement in continuous time. In this paper we review the link between these two descriptions and we study the properties of the output of the measurement. For simplicity we deal only with the diffusive case. Firstly, we discuss the quantum stochastic Schrödinger equation, which is based on quantum stochastic calculus, and we show how to transform it into the classical stochastic Schrödinger equation by diagonalization of suitable commuting quantum observables. Then, we give the a posteriori state, the conditional system state at time given the output up to that time, and we link its evolution to the classical stochastic Schrödinger equation. Moreover, the relation with quantum filtering theory is shortly discussed. Finally, we study the output of the continuous measurement, which is a stochastic process with probability distribution given by the rules of quantum mechanics. When the output process is stationary, at least in the long run, the spectrum of the process can be introduced and its properties studied. In particular we show how the Heisenberg uncertainty relations give rise to characteristic bounds on the possible spectra and we discuss how this is related to the typical quantum phenomenon of squeezing. We use a simple quantum system, a two-level atom stimulated by a laser, to discuss the differences between homodyne and heterodyne detection and to explicitly show squeezing and anti-squeezing of the homodyne spectrum and the Mollow triplet in the fluorescence spectrum.
Also at ]Istituto Nazionale di Fisica Nucleare, Sezione di Milano, and Istituto Nazionale di Alta Matematica, GNAMPA
A big achievement in the 70’s-80’s was to show that, inside the modern formulation of quantum mechanics, based on positive operator valued measures and instruments Kra80 (); Dav76 (), a consistent theory of measurements in continuous time (quantum continuous measurements) was possible Dav76 (); BarLP82 (); BarL85 (); Bar86PR (); Bel88 (); BarB91 (); Hol01 (). Starting from the 80’s, two other very flexible and powerful formulations of continuous measurement theory were developed. The first one is often referred to as quantum trajectory theory and it is based on the the stochastic Schrödinger equation (SSE), a stochastic differential equation of classical type (commuting noises, Itô calculus) Bel88 (); BarB91 (); Car93 (); WisMil93 (); BarP96 (); WisM93 (); Wis96 (); Mil96 (); Car08 (); BarG09 (); WisM10 (). The second formulation is based on quantum stochastic calculus HudP84 (); GarC85 (); Parthas92 () and the quantum SSE (non commuting noises, Bose fields, Hudson-Parthasarathy equation) BarL85 (); Bar86PR (); Bel88 (); BarP96 (); GarZ00 (); Hol01 (); Bar06 (); Car08 (); WisM10 (). The main applications of quantum continuous measurements are in the photon detection theory in quantum optics (direct, heterodyne, homodyne detection) Bar90QO (); Car93 (); WisMil93 (); WisM93 (); BarP96 (); Wis96 (); Mil96 (); GarZ00 (); BarL00 (); BarP02 (); BarG09 (); Bar06 (); Car08 (); WisM10 (). While the classical SSE gives a differential description of the joint evolution of the observed signal and of the measured system, in agreement with the axiomatic formulation of quantum mechanics, the quantum SSE gives a dilation of the measurement process, explicitly introducing an environment which interacts with the system and mediates the observations.
In this paper we start by giving a short presentation of continuous measurement theory based on the quantum SSE and we show the equivalence between this approach and the one based on the classical SSE (Secs. II and III). Then we consider the output of a continuous measurement and we develop the theory up to the introduction of its spectrum (Sec. IV), which enables the study of typical and significative applications, see Sec. V.
We consider only the type of observables relevant for the description of homodyne/heterodyne detection and we make the mathematical simplification of introducing only bounded operators on the Hilbert space of the quantum system of interest and a finite number of noises; for the case of unbounded operators see FagW03 (); Fag06 (); CastroB11 ().
In Sec. II, first we discuss some typical approximations that give rise to a system/environment interaction described by a quantum SSE, then we give a mathematical meaning to such an equation by introducing the basic ingredients of quantum stochastic calculus on Fock space.
In Sec. III we introduce the quantum observables which describe the continuous measurement and we show how to derive the classical SSE and the related stochastic master equation (SME). The key point in the step from the quantum SSE to the classical SSE is the introduction of an Hilbert space isomorphism which diagonalizes a suitable complete set of quantum observables. We shortly illustrate also the connections of this approach to quantum filtering theory Bel88 (); BouVHJ07 (). The classical SSE and the SME give both the probability distribution for the observed output and the a posteriori state, the conditional system state given a realization of the output. These equations are driven by classical noises, but, in spite of this, they are fully quantum as they are equivalent to the formulation of continuous measurements based on quantum fields. It is just in the formulation based on SSE and SME that the probabilistic structure of the output current becomes very transparent.
In Sec. IV we introduce the spectrum of the classical stochastic process which represents the output and we study the general properties of the spectra of such processes by proving characteristic bounds due to the Heisenberg uncertainty principle. This bound is one of the evidences that the whole theory of continuous measurements is fully quantum, independently of the adopted formulation.
As an application, in Sec. V we present the case of a two-level atom, which is measured in continuous time by detection of its fluorescence light. The spectral analysis of the output can reveal the phenomenon of squeezing of the fluorescence light, a phenomenon related to the uncertainty relations. We use this example also to illustrate the differences between homodyning and heterodyning and between the spectrum of the squeezing and the power spectrum. Finally we show how Mollow triplet appears in the power spectrum in the case of an intense stimulating laser. Section VI contains our conclusions.
Ii The quantum stochastic Schrödinger equation
We want to introduce the continuous measurement theory of a system at its higher level, that is by explicitly modelling also the quantum environment which mediates the observation: we observe an environment that is coupled with , thus acquiring direct information on the environment and indirect information on .
Thus we start from the quantum SSE which will be used to define the global evolution of and the environment. Even if not defined by a regular Schrödinger equation, such an evolution will be a proper Hamiltonian evolution, but of a very particular kind, with “Markovian” features for . Indeed, the quantum SSE naturally emerges with the typical “Markovian” limits which allow to describe an open evolution of by a quantum dynamical semigroup or to describe a continuous measurement of by an instrumental process Hol01 ().
Therefore a quantum SSE defines an approximated model, which, nevertheless, is fully quantum, and for this reason it can describe very well typical quantum experiments, such as the quantum optical ones or in general quantum continuous measurements.
In this section, first we show some typical approximations which lead to a quantum SSE, then we introduce quantum stochastic calculus (QSC), the mathematical background which gives a meaning to that equation and allows to start with its study.
In the following we shall denote by the system space, the complex separable Hilbert space associated to the open quantum system .
ii.1 The physical bases of the quantum SSE
Our system interacts with some quantum Bose fields , satisfying the the canonical commutation relations (CCR)
The parameter is the energy or the frequency (we are taking ) of the free field, while is an additional discrete degree of freedom. These fields can represent, for instance, the electromagnetic field; in this case the index stays for polarization, (discretized) direction of propagation, and so on Bar97 ().
A generic system–field interaction, linear in the field operators, can be written as
where the are system operators (acting on ) and the are real couplings. In the optical case, typically the are dipole operators and the rotating-wave approximation is understood. The are resonance frequencies of system and is the interaction bandwidth.
By working with the Heisenberg equations of motion for system operators, Gardiner and Collet GarC85 () discussed the approximations needed to pass from the quasi–physical Hamiltonian (2) to a Markovian quantum stochastic evolution. Here we present the same approximations by working with the global unitary evolution of the composed system “ plus fields” Bar90QO ().
The flat-spectrum approximation
The first approximation is to take the couplings independent from : the flat-spectrum approximation. As a constant can always be included in , we take . Then, we pass to the interaction picture with respect to the free dynamics of the fields:
In this picture, the interaction Hamiltonian becomes
By construction, the field operator represents a wave packet with some carrier frequency and bandwidth .
In the interaction picture, the time evolution operator can be written as
where is the free Hamiltonian of system and is the usual time–ordering prescription (chronological product).
The broad-band approximation
Now, we take the broad-band approximation: , . Note that in this limit the energy of the free field becomes unbounded both from above and from below. This approximation is justified when only energies not far from the resonance frequencies are involved in the physical process (GarZ00, , p. 149).
and satisfy the CCR
Then the free dynamics of the fields (3) gives
so that the argument in the fields has a double role: it is a field degree of freedom, the conjugate momentum of the free field energy , because of (7), and it is the time, because is the evolution of at time . Let us note that Bose fields with delta-commutations in time were already found by Yuen and Shapiro YuenS78 () in their study of the quasi-monochromatic paraxial approximation of the electromagnetic field.
Let us take now the limit , , in the Hamiltonian and in the evolution . Formally
which describes a singular interaction, but which is not a proper operator in a Hilbert space because the singular field operator is not integrated over . Nevertheless, where
which can be a proper unitary evolution in a Hilbert space thanks to QSC. Indeed, QSC is the mathematical theory that gives a meaning to Itô-type integrals with respect to the non-commuting noises and , that is with respect to (GarZ00, , Eq. (11.2.24))
To find an equation for we can write
Due to this fact, in the second order term of the expansion of the exponential surely some of the contributions are of order . This shows that new mathematical tools are needed to treat the singular interaction appearing in (10), QSC indeed. In addition, here another peculiarity arises: in the case of quantum fields there exist non unitarily equivalent representations of the CCR (8). Moreover, which ones of the second order terms are indeed of order depends on the representation and this implies that the rules of QSC depend on the representation of the CCR HudL85 ().
In this paper we consider only the representation of the CCR (8) on the Fock space, the one characterized by the existence of the vacuum state. Let us stress that representations not unitarily equivalent to the Fock one describe physically different situations, such as thermal and squeezed input fields GarC85 (); GarZ00 ().
The final result of these approximations is the quantum SSE (22), as we shall see after Theorem 1. Note that, however, such an equation is a general evolution model that emerges under many different Markovian limits, not only with the one we have just described. Of course, the system operators and in (10) or (22) are to be chosen just by looking at the physical context and the approximations that produce the Markovian regime. In the model of Sec. V we shall show how to represent various dissipative effects by a suitable choice of the operators .
ii.2 Quantum stochastic calculus and unitary dynamics
We introduce now QSC and the Hudson-Parthasarathy equation in the Fock representation. QSC HudP84 () is based on the use of some Bose fields, satisfying the CCR with a Dirac delta in time (8), that model the environment interacting with an initial system and play the role of non-commuting noises. By QSC one gives meaning to the quantum stochastic Schrödinger equation or Hudson-Parthasarathy equation HudP84 (); Parthas92 (). For a short review see (Bar06, , Sec. 2) or (GarZ00, , Secs. 11.1, 11.2). Our aim is to recall the main notions and to fix the notations, not to give a self-contained presentation, which can be found in Parthas92 (); Bar06 ().
Let be the separable Hilbert space of a bosonic particle and be the “-particle space”, that is the symmetric part of the tensor product , times. Then, the direct sum is the symmetric (or bosonic) Fock space over . In this context a coherent vector , , is the vector in given by
Note that represents the vacuum state and that ; in particular, the coherent vectors are normalized. Moreover, they are all linearly independent and their linear combinations are dense in . Then, an important property of the Fock spaces is that the action on the coherent vectors uniquely determines a densely defined linear operator. We are interested in and we denote by the symmetric Fock space over the one-particle space .
Factorization properties of the Fock space
A general property of symmetric Fock spaces is that, when the one-particle space is given by a direct sum (), then the factorization property holds.
In our set up, for every time interval , let us denote by the symmetric Fock space over ; in particular, we have . Then, for any , we have and
Moreover, each space can be identified with a subspace of the full Fock space by taking the tensor product of a generic vector in with the vacuum of . Then, for every , we have the identification
We are denoting by the indicator function of the set and by the restriction of the function to the set . With an abuse of notation we write
In particular, can represent a vector in or in and we have the identification .
Let be the canonical basis in and for any let us set .
Then, we define two families of mutually adjoint operators, the annihilation and creation processes, by their actions on the coherent vectors:
The overline denotes the complex conjugation.
For , the annihilation and creation processes are adapted, in the sense that they factorizes, with respect to (14), as
and they satisfy the integrated form of the CCR, namely
is the minimum between and and is the identity operator on .
Temporal modes and Weyl operators.
The free evolution in the Fock space is represented by the left shift in
Then, coherently with (9), the action of the shift on the fields is given by
wich is the free evolution (9).
If we take a function we can define the annihilation operator
By Eq. (16), its action on the coherent vectors is given by
If we take a complete orthonormal system , , in , we can define the annihilation operators . Together with their adjoint operators, they satisfy the usual CCR. We can say that the upper index denotes the temporal modes, while the lower index denotes the polarization/spatial modes.
An important technical tool is represented by the Weyl operators , , the unitary operators defined by: ,
this is nothing but the displacement operator for the field. By using the notation (18) we can write
while, by using the discrete modes introduced above, we have
By h.c. we denote the Hermitian conjugate operator.
The Hudson-Parthasarathy equation
Now we want to couple the system with the fields by constructing a unitary evolution of the composite system in . When convenient, an operator on (resp. on ) is identified with on (resp. ).
By defining integrals of Itô type with respect to the increments of the quantum processes , , it is possible to construct adapted operator processes on and to develop a quantum stochastic calculus, whose rules are summarized, at a heuristic level, by the quantum Itô table
Let us stress that these multiplication rules do not depend on the field state in the Fock space, but they depend on the representation. Indeed, thermal and squeezed representations, which describe different physical situations, have different Itô tables (GarZ00, , Eqs. (5.3.52), (10.2.38)).
Theorem 1 (Hudson and Parthasarathy).
Let , , , be bounded operators on such that . We set also
Then, the quantum stochastic differential equation
with the initial condition , has a unique solution, which is a strongly continuous adapted family of unitary operators on . Moreover, the family of unitary operators , , and , , is a strongly continuous unitary group.
Note that, if we take our limit dynamics (10) of Sec. II.1, and we compute the differential in expression (12) by expanding the exponential with the Itô table (20), we get indeed Eq. (22). So, the unitary operators represent the system-field dynamics in the interaction picture with respect to the free field evolution.
Then, for , the dynamics in the Schrödinger picture is the unitary group , whose Hamiltonian is a singular perturbation of the unbounded generator of Greg00 (); Greg01 (). Roughly speaking, the system absorbs or emits bosons instantaneously; then, the emitted bosons are carried away by their free dynamics and never come back.
Note that the interaction picture with respect to the free field dynamics coincides with the Schrödinger picture when only reduced system states and observables are considered.
ii.3 The reduced dynamics of the system
The states of a quantum system are represented by statistical operators, positive trace-class operators with trace one; let us denote by the set of statistical operators on . For every composed state in , the partial trace (resp. ) with respect to the field (resp. system) Hilbert space gives the reduced system (resp. field) state in (resp. in ).
The initial state and the reduced states
As initial state of the composed system “ plus fields” we take , where is generic and is a coherent state, . Then, the system-field state at time , in the field interaction picture, is
The reduced system state and the reduced field state are
The master equation
One of the main properties of the Hudson-Parthasarathy equation is that, with the initial state introduced above, the reduced dynamics of system exactly obeys a quantum master equation HudP84 (); Parthas92 (); Bar06 (). Indeed, we get
where the Liouville operator turns out to be
Therefore, is an open system, as it interacts with the fields in , and its evolution turns out to be Markovian thanks to the properties of the interaction and of the choice of a coherent state as initial state of the environment. Note that the dynamics (25) depends not only on the global evolution (22) but also on the initial state of the environment .
It is useful to introduce also the evolution operator from to by
With this notation we have .
Iii Continuous monitoring
The connections among quantum stochastic calculus, quantum Langevin equations and input and output fields were developed by Gardiner and Collet in GarC85 (). Then, in Bar86PR () these notions were connected to the unitary evolution (22) and to continuous measurements. Indeed, another fundamental property of the Hudson-Parthasarathy equation is that it allows for a fully quantum description of a continuous measurement of the system : the measurement is obtained by detecting the bosons that have been emitted by . Of course such a measurement acquires information on both and the detected bosons.
iii.1 Input and output fields
Let us call “input fields” the fields , when they are considered as operators in interaction picture at time , with respect to , and let us call “output fields” the same fields in the Heisenberg picture:
and a similar definition for . By the properties of the Fock space and of the unitary operators , it is possible to prove that
This equation is of fundamental importance and it immediately implies that the output fields satisfy the same commutation rules of the input fields, for instance the CCR (15): the output fields remain Bose free fields. By applying the formal rules of QSC (20), we can express the output fields as the quantum stochastic integrals Bar86PR ()
is given by the adjoint expression.
iii.2 The field observables
The key point of the theory of continuous measurements is to consider field observables represented by time dependent, commuting selfadjoint operators in the Heisenberg picture BarL85 (); Bar86PR (); Bar06 (). Being commuting at different times, these observables represent outputs produced at different times which can be obtained in the same experiment. Here, the observables we consider are some field quadratures. Let us start by introducing the selfadjoint operators
the phase and the function , with , are fixed.
The operators (32) have to be interpreted as linear combinations of the formal increments , which represent field operators in the interaction picture. The corresponding operators in the Heisenberg picture are
where the second equality follows from Eq. (30). These “output” quadratures are our observables.
When “field 1” represents the electromagnetic field, a physical realization of a measurement of the observables (33) is implemented by what is called balanced heterodyne/homodyne detection ShYM79 (); YuS80 (); YuenC83 (), (GarZ00, , Sec. 8.4.4). The light emitted by the system in the “channel 1” interferes with an intense laser beam represented by the wave , the local oscillator; as , it represents only the phase of the local oscillator wave. The description of the apparatus and its formalization in mathematical terms is given in (Bar06, , Sec. 3.5).
Each quadrature is observed at the corresponding time and it regards those bosons in “field 1” which have eventually interacted with between time and time , so it can be interpreted as an indirect measurement performed on the system .
Therefore, the observables , , can be jointly measured for every interaction (22). The output is a (random) number at every time , that is a signal depending on time, a stochastic process, which is the result of a continuous indirect monitoring of the system . Its probability distribution is given by the usual postulates of quantum mechanics trough the joint diagonalization of the operators . Actually, always thanks to Eq. (33), it will be enough to jointly diagonalize the operators .
Let us stress that quadratures of type (32) with different phases and functions represent incompatible observables, because they do not commute but satisfy
Note that for we get
and for they commute as anticipated.
iii.3 The stochastic representation
The commuting selfadjoint operators (32) have a joint projection valued measure (pvm) ; by Born rule, it gives the probability distribution for the output of the continuous measurement. Moreover, via the partial trace on the fields, gives also the instruments describing the transformations of from time 0 to an arbitrary time , conditioned on the information acquired up to time . Furthermore, via joint diagonalization and conditioning, the pvm even gives the stochastic evolution of the conditional state (or a posteriori state), the state of at time given the observed signal from time 0 to time . This evolution turns out to satisfy a stochastic differential equation (SSE or SME), with classical driving noises. The introduction of such stochastic evolution equations for the conditional state was an achievement of the quantum filtering theory Bel88 (); BelS92 (); Bel89 (); Bel94 (); BouVHJ07 (); BarB91 ().
The passage from the formulation with quantum fields and Hudson-Parthasarathy equation to the one based on classical stochastic differential equations can be done by different techniques. The technique based on the use of isomorphisms between the Fock space and the Wiener space is very powerful and clear; here we present a variant of the construction given in BarP96 ().
Let us note that the observation we consider is not complete, because it regards only field 1 and involves only positive times. To make unique the isomorphism which diagonalizes the self-adjoint operators (32), we need to add fictitious observations, involving quadratures of the fields too. So, we take a function such that
Then, we introduce the field quadratures: for ,
We use this definition for positive and negative times by taking the convention for a negative . These quadratures form a complete set of compatible observables on the Fock space . Note that . In the following subsection we jointly diagonalize all the observables (38) by introducing an explicit isomorphism between Fock and Wiener spaces.
Spectral representation on the Wiener space
Fixed the functions , that is the field quadratures (38), we look for a probability space , a unitary operator (the Hilbert space of the complex square integrable random variables on the given probability space), and a family of random variables on such that
for all , , for almost all , and for all in the domain of the selfadjoint operator . This means that each is represented in as the multiplication operator by . We can get such a joint diagonalization on the space of the canonical representation of the Wiener process; a short presentation of the canonical Wiener process is given in (BarG09, , Secs. A.2.4, A.2.6).
Remark 1 (The Wiener space).
Let be the space of the continuous functions such that . We define the -dimensional process , , by and we denote by the smallest -algebra of subsets of for which these functions are measurable: . Then, there exists a unique probability measure on the measurable space , the Wiener measure, such that the processes , , , are independent standard Wiener processes. Moreover, for positive times we introduce the natural filtration of the process : . Finally, the Hilbert space is called Wiener space.
Let us also recall that, if , then their inner product is given by the -expectation :
Definition 1 (The isomorphism ).
Let be the linear operator defined by: ,
In particular we have
The operator turns out to be an isomorphism and it realizes the representation (39): , i.e. the field quadratures are mapped into the operators “multiplication by the Wiener processes”. Because the isomorphism jointly diagonalizes all the observables (38), then their joint pvm on the Fock space is , .
The distribution of the output Let us restrict now to the observed quadrature (32); the -algebra is the space of all the events regarding our observables , . Then, the joint pvm of the observed quadratures is defined on the measurable space by
Finally, we get the distribution of the output. By setting , then, , is the space of the observed events up to time , associated to the observables for times from 0 to , and, according to the usual rules of quantum mechanics, the probabilities of such events are given by
Note that, when the field state is the vacuum and there is no interaction between system and the fields, this probability reduces to . This means that in this case the quadratures (32) are distributed as a standard Wiener process.
Let us stress that the pvm (41) depends on the parameters and defining the quadrature (32); these parameters are contained in the definition of the isomorphism (40). On the contrary, the choice of the trajectory space (the measurable space ) and the definition of are independent of and . With respect to the time dependence, the physical probabilities (42) are consistent, i.e.
A more detailed study of the statistical properties of the output needs the introduction of the characteristic operator (Sec. III.5).
The observation of the emitted field can be interpreted as an indirect measurement on the system and this is formalized by the concept of instrument Kra80 (); Dav76 (). The family of instruments , , describing our measure is defined by: , ,
For , the initial system state, Eq. (44) gives the unnormalized state of at time conditioned on the information that the values of the signal in the time interval from 0 to were in . Of course we have
while the normalized conditioned state is given by divided by its trace (45).
Let us remark that
so that the system reduced state at time in the case of no observation () coincides with the so called a priori state (), that is the system state at time in the case of observation performed but not taken into account. This is in agreement with our rough picture of the measurement process: we observe fields which have already interacted with system and which will never interact again with it. This means that we acquire information on , as we have , but we do not add any perturbation on its evolution as we have .
The a posteriori states Now we want to introduce , the state of at time conditioned on the whole information supplied by our indirect measurement between time 0 and time , that is by the signal produced by the measurement of for . Therefore, has to be a random state depending on the output , , that is a random state measurable with respect to ; in other terms, we have the functional dependence . Such a state is called a posteriori state and it is determined by the initial state and by the instrument : it is the unique -measurable random state such that
The definition of a posteriori state is not linked only to measurements in continuous time, but it has been introduced for a generic instrument Ozawa ().
As we have a reference probability on the output space we can equivalently look for the unnormalized a posteriori state , the unique -measurable random positive operator such that
Then, is the probability density of with respect to and we have .
The unnormalized a posteriori state can be computed by using the spectral representation (39) of the operators and its evolution can be obtained by passing through the SSE.
The stochastic Schrödinger equation
In order to compute the a posteriori state of our instrument (44), it is convenient to pass through two fictitious instruments: , associated to a complete set of compatible observables in , and , associated to a complete set of compatible observables in . This latter instrument has the simple a posteriori state (51), whose evolution is given by the SSE (53).
First of all, let us imagine, in the Heisenberg picture, that in the time interval we measure all the quadratures , , , and moreover we conclude the measure by observing at time also the field observables , and or . This is a family of commuting observables, thanks to (35) and to (30), that implies for every . Then, the instrument associated to this fictitious measurement is given by an expression analogous to (44). Using again the joint pvm , , of the quadratures , if the system initial state is pure, , , , then ,