FACOLT `A DI SCIENZE MATEMATICHE, FISICHE E NATURALI Corso di Laurea Magistrale in Fisica
Tesi di laurea magistrale
Continuous-measurement as a tool for
squeezing generation and dynamic
stabilization of bosonic quantum systems
Candidata:
Jinglei Zhang
Relatore:
Dott. Marco G. Genoni
Correlatore:
Dott. Alessio Serafini
In Quantum Mechanics (QM) measurements of observables are traditionally introduced as instantaneous, but experimentally this assumption is often simply not true. So a theory for measurements performed continuously in time has been developed and its applications have became increasingly im-portant mainly due to the growing interest in the application of feedback control in quantum systems, especially in quantum information and compu-tation for continuous variable systems.
This Thesis develops applications of the theory for continuous measure-ment to experimeasure-mentally feasible bosonic systems and explores the resulting advantages.
The first chapter of this Thesis consists of a review about basic facts in QM. Because the main subject is continuous-measurements, measurement theory is treated with particular attention: besides postulating projective measurements, the theory for postive-operator valued measurements is de-veloped. In order to set the scenario for the applications chapter, in the second part of chapter 1 Gaussian states for bosonic systems are introduced. The importance of Gaussian states relies in the fact that they are easily generated in laboratory with quantum optics and bilinear interaction Hamil-tonians and as such, are a commonly studied quantum systems. So the main characteristics of Gaussian states and the unitary evolution generated by bilinear Hamiltonians are presented.
The second chapter develops the theory of continuous-measurement with Markovian feedback. Firstly, in section 2.1, the general form of a Markovian master equation describing an open quantum system is presented. This is followed by an explicit example of a single bosonic mode interacting with a bosonic Markovian bath. In section 1.1.4 the notions introduced in the first chapter are used to study the details of two examples of quantum measure-ment for bosonic quantum system that are experimeasure-mentally very important, namely the photodetection and the homodyne detection. Finally in section 2.3 we study continuous-measurements starting with two examples in order to present the problem in a physically concrete setting. In the first case a quantum system subjected to the continuous monitoring of the position ob-servable is described. In this simpler case we also study the effects of multiple and inefficient observers. In the second case we derive the stochastic master
equation for a bosonic mode interacting with an environment which is contin-uously monitored through a homodyne detection. In section 2.3.3, we present the general mathematical formalism describing continuous-measurement and feedback control for linear quantum system. This formalism allows to intro-duce important results and features of continuous-measurements. In partic-ular we focus on the stabilization of unstable systems by means of continuous monitoring, the possibility to stabilize the stochastic dynamics of Gaussian states and to generate steady-state squeezing and entanglement, by using a simple Markovian feedback protocol.
The third chapter contains the original results of the thesis. Here we focus on specific bosonic systems and we investigate the possibilities offered by continuous measurement in order to deal with unstable dynamics and to generate squeezed states, i.e. states with fluctuations below the vacuum level, that are considered resources both for quantum information and quantum metrology.
In the first case we study a degenerate parametric amplifier. It is a commonly used optical device to generate squeezed states of light. It is known in literature that there exist a3 dB limit on the achievable squeezing for a steady-state, but in the previous chapter it is shown that, in the case of a continuously monitored system, a new ultimate bound exists. We study in detail the evolution of the parametric amplifier when continuous homodyne detection is performed and we also show that the new optimal bound can actually be achieved when the system is monitored with maximum efficiency. We also study the achievable steady-state squeezing for a generic efficiency and calculate the relation between the physical parameters in order to obtain a squeezing amplification larger than the old3 dB bound.
Eventually an opto-mechanical system is considered: a nano-dielectric trapped in a harmonic potential by optical tweezers and interacting with a single cavity mode. This is an experimentally achievable system and several research groups are working on the possibility to cool the centre of mass mo-tion of the dielectric to its quantum ground state in the harmonic potential that confines it. We thus have a two-mode system consisting of a mechanical oscillator and a single cavity mode and two possible mode on which to apply continuous measurement. Studying the corresponding master equation, it is possible to show that the dynamics of the system is not stable, due to the decoherence introduced by the mechanical oscillator and therefore there is not a steady-state. In this thesis we prove that a monitoring with any efficiency on any mode is sufficient to generate a stable dynamics. In par-ticular, when a continuous measurement on the position of the mechanical oscillator is performed, the purity of the dielectric is studied when varying the efficiency of the measurement and it is shown that a pure state is ob-tained for maximum efficiency. It is also shown that there exist a minimum value for the efficiency in order to obtain sub-shot noise squeezing for the me-chanical oscillator. Finally we explore how the variations of the oscillator’s
frequency and the coupling strength between the mechanical and the optical mode change the achievable squeezing for the oscillator, in order to provide indications for future experimental improvement of the implementation of the system.
1 Introduction 1
1.1 Introduction to Quantum Mechanics . . . 1
1.1.1 State space . . . 1 1.1.2 Composite systems . . . 1 1.1.3 Density operator . . . 2 1.1.4 Time evolution . . . 2 1.1.5 Measurement theory . . . 4 1.1.6 Generalized measurements . . . 7 1.2 Bosonic systems . . . 8
1.2.1 Characteristic function and Wigner function . . . 11
1.2.2 Symplectic transformation . . . 12
1.2.3 Linear and bilinear interaction of modes . . . 13
1.2.4 Definition of Gaussian states . . . 17
2 Open quantum systems and continuous-measurement 22 2.1 Markovian master equation . . . 22
2.1.1 An example: the bosonic lossy channel . . . 23
2.2 Quantum measurement of continuous-variable systems . . . . 26
2.2.1 Photodetection . . . 26
2.2.2 Homodyne detection . . . 27
2.3 Continuous measurement and SME . . . 29
2.3.1 SME for continuous position monitoring . . . 30
2.3.2 SME for continuous homodyne detection . . . 35
2.3.3 Continuous-measurement and optimal unravellings for feedback control in linear quantum systems . . . 38
3 Applications of continuous measurement 43 3.1 Quantum parametric amplifier . . . 43
3.1.1 The physics of the quantum parametric amplifier . . . 43
3.1.2 Continuous monitoring of a parametric amplifier . . . 45
3.2 Opto-mechanics with a levitating bead . . . 50
3.2.1 The physics of the opto-mechanical levitating bead . . 50
3.2.2 Continuous monitoring of the levitating bead . . . 53
4 Conclusions and outlooks 60
A Derivation of the Markovian master equation 62
B Stochastic differential equations 66
B.1 Connection to the Fokker-Planck equation . . . 68
C Linear classical systems 69
C.0.1 Stabilizability and detectability . . . 70
Introduction
1.1
Introduction to Quantum Mechanics
This section gives the basic facts about Quantum Mechanics (QM) as a math-ematical framework in which to develop a physical theory. The postulate of QM will be stated followed by a brief discussion.
1.1.1 State space
The first postulate states that to every physical system is associated a com-plex vector space with inner product, that is a Hilbert space H. The state of the system that gives our knowledge about it is a unit vector in H called the state vector.
The simplest Hilbert space is two-dimensional and is associated to a so-called qubit. Given an orthonormal basis {|0i , |1i}, the most general state vector is |ψi= a |0i + b |1i with a, b, ∈ C. The normalization condition states that |a|2+ |b|2 = 1.
The dimension of H can also be infinite. The main object of study of this thesis will be indeed systems with an infinite, but numerable orthonormal basis.
1.1.2 Composite systems
The following postulate states how to describe the state of multiple systems considered together.
Given two systems A and B associated respectively to HAand HB, the
Hilbert space associated with the composite system is the tensor product HAB = HA⊗ HB.
1.1.3 Density operator
If a system has probability pj of being in the state |ψij for j = 1, . . . , N , it
is possible to define the density operator of the system as ρ=
N
X
j=1
pj|ψjihψj| . (1.1)
The density operator is a way to describe a system when there is uncertainty about the precise state vector. A simple state vector is called a pure state, while a density operator describes in general a mixed state when N ≥ 2. Every density operator satisfies the following:
Properties 1.
ρ ≥0, Tr [ρ] = 1. (1.2)
The set {pj, |ψji} is called an unravelling for the density operator. For
pure states there is only one possible unravelling, but this is not true for a generic mixed state.
Theorem 2. Two unravellings {pj, |ψji} and {qj, |φji} correspond to the
same density operator if and only if there exist an unitary matrix U such that √ pj|ψji = X i Uji √ qi|φii . (1.3)
Properties 3. The set of density operator is convex. That is, if ρ and σ are density operators than their convex linear combination pρ+ (1 − p)σ is still a valid density operator.
1.1.4 Time evolution
Time evolution for pure states
Given a state vector |ψi of a closed physical system (that is non interacting with any other system), its time evolution is described by an unitary operator
|ψ(t)i = ˆU(t) |ψ(t = 0)i . (1.4)
The unitary operator is generated by a self-adjoint operator ˆH called Hamil-tonian of the system, meaning that ˆU(t) = exph−i ˆHt
i
(~ = 1 is assumed). In terms of the Hamiltonian, the evolution is described bySchrödinger’s equa-tion:
d|ψi
dt = H |ψi . (1.5)
Sometimes it is possible to describe the evolution of an open system with time-dependent Hamiltonian, for example studying an atom in an atom-laser system (Ref. [NC10]), but other times the evolution of an open system will simply not be unitary, for example when a system is measured by an observer.
Time evolution for density operators
The most general evolution for a density operator is given in terms of a super-operator mapΦt such that
ρ(t) = Φtρ(t = 0). (1.6)
In the case of a unitary evolution (as the one described above), the super-operator corresponds to
Φt(ρ) = ˆU ρ ˆU†. (1.7)
In general, the only requirement onΦt is that it must transform density
operators in density operators. Hence, a generic map Φt must satisfy the
following properties: 1. Φ is linear;
2. Φ is trace preserving (TP), that is Tr [Φ(Θ)] = Tr [Θ], where Θ is a generic matrix, this is necessary sinceTr [ρ] = 1 for all density opera-tors;
3. Φ is positive, that is if Θ ≥ 0, then Φ(Θ) ≥ 0, this is necessary because all density operators are positive defined;
4. Φ is completely positive (CP), this means that Φ is positive and Φ ⊗ 1 is also positive, where 1 acts on any ancillary system.
The last property makes safe the case of a system interacting with an ancil-lary system, after applying the super-operatorΦ ⊗ 1, the reduced matrix of the system remains a sensible (positive defined) density operator.
The super-operators that satisfy all these properties are called completely-positive trace-preserving maps (CPTP maps) or quantum channels.
A typical example of a positive that is not completely positive is the transposition map: T(Θ) = ΘT. It can be shown that this map satisfies
properties from 1. to 3., but is not completely positive.
Let us derive the super-operator for the case of time evolution for a system S interacting with the environment E. The global system is closed and evolve through the unitary operator ˆUSE(t). Let {|li} be an orthonormal
basis for HS and let the initial state of the system be
ρSE(0) = ρS(0) ⊗ |0iEh0| . (1.8)
After the time evolution the global state is
but if we only care about S, its final state will be ρS(t) = TrE[ρSE(t)] = = TrEh ˆUSE(ρS(0) ⊗ |0iEh0|) ˆUSE† i = =X l hl| ˆUSE|0i E EρS h0| ˆU † SE|li E E = =X l ˆ MlρSMˆl†, (1.10)
where we defined the Kraus operator ˆMl≡ Ehl| ˆUSE|0iE on HS.
It can be shown that the maps defined by Θ 7→ Φ(Θ) =X
l
ˆ
MlΘ ˆMl† (1.11)
is indeed CP, and the trace-preserving property is ensured by the fact that these Kraus operators satisfy
X
l
ˆ
MlMˆl†= ˆ1S. (1.12)
On the other hand, given a set of operatorsn ˆMl
o
on H that satisfy the normalization condition (1.12), there are infinitely many ways to find and ancillary system and a global unitary evolution that generates these Kraus operators.
1.1.5 Measurement theory
The traditional approach to quantum measurement is to postulate the pro-jective measurements. It can be described as follows: consider the state of a quantum system described by the density matrix ρ, and an operator ˆΛ (also called observable) that can be diagonalized as
ˆ
Λ =X
λ
λ ˆΠλ, (1.13)
where the λ are the eigenvalues of Λ, which are real and assumed to be discrete for convenience. The ˆΠλ are the projector onto the subspace of the
eigenvectors with eigenvalues λ. As a consequence of the spectral theorem, these projectors are orthonormal, that is
ˆ
ΠλΠˆ0λ = δλ,λ0Πˆλ. (1.14)
• If the spectrum {λ} is non degenerate, the Πλ are rank-1 projector
ˆ
Πλ = |λi hλ| and the measurement is also called von Neumann
• If one of the eigenvalues λ is Nλ-fold degenerate, then the projector
onto its subspace has rank Nλ and can be written as
ˆ Πλ = Nλ X j=1 |λ, jihλ, j| . (1.15)
When the operator Λ is measured, one of the eigenvalues λ is obtained with probability
Pr(λ) = Tr[ρ ˆΠλ], (1.16)
and the projection postulate states that immediately after the measurement, the state of the system is
ρ0 = Πˆλρ ˆΠλ Tr[ρ ˆΠλ]
. (1.17)
The evolution from ρ to ρ0 is called state collapse; it is instantaneous and non linear and its interpretation remains a puzzling question that deals with the fundamentals of Quantum Mechanics but will not be addressed in this thesis.
If the system is in a pure state ρ = |ψi hψ| the analogous of equa-tions (1.16) and (1.17) are respectively
Pr(λ) = hψ| ˆΠλ|ψi |ψi0= ˆ Πλ|ψi q hψ| ˆΠλ|ψi (1.18)
Conditional and unconditional states
For the sake of clarity, the state of the system ρ0 after the measurement will be called conditional state, meaning that the state is conditioned by the mea-surement performed and the knowledge of the outcome of the meamea-surement. From now on the conditional state with outcome λ will be labelled with the subscript c as in ρc(λ) (λ sometimes will be omitted).
The change in the state of the system due to the measurement can be interpreted as the change of the observer’s knowledge about that system and depends on the new information acquired.
One can also make the measurement and forget the outcome, then the state of the system after the measurement is the weighted average of the possible conditional states:
ρunc= X λ Pr(λ)ρc(λ) = X λ ˆ Πλρ ˆΠλ. (1.19)
Reduced density operator
Consider two systems A, B with their respective orthonormal basis {|iiA} and {|kiB} and their global Hilbert space HAB = HA ⊗ HB. The most
generic pure state of the global system can be written as |ψiAB =X
i,k
αik|iiA⊗ |kiB. (1.20)
If ˆΛAis an observable on the system A, the corresponding observable on the
global system is ˆΛA⊗ ˆ1B. The mean value of ˆΛA is
h ˆΛAi = ABhψ| ˆΛA⊗ ˆ1B|ψiAB = X i,j,k α∗jkαikAhj| ˆΛA|iiA≡ TrAh ˆΛAρA i , (1.21) where the reduced density matrix of A is defined to be
ρA=
X
i,j,k
α∗jkαik|iiAhj| . (1.22)
Thus the density operator that we have introduced before in section 1.1.3 does not need a new postulate: physically the uncertainty about the precise state of the system can be thought as due to the ignorance about the cor-relation of the system with the environment. Indeed mathematically, given the density operator ρS of a system, it is always possible to construct a Hilbert space corresponding to some ancillary system so that ρS is the re-duced density matrix of a pure state of the global system. This process is called purification of ρS and there are infinitely different ways to purify a density operator.
Properties 4. Consider a system A with density operator ρA and unravel-lingpj, |ψjiA
j=1,...,d. There exist a d-dimensional Hilbert space HB and a
pure space |ψiAB in the global Hilbert space Ha⊗ HB such that
ρa= TrB[|ψiABhψ|]. (1.23)
Proof. Considering an orthonormal basis {|eji}j=1,...,d, let us define the global
system’s vector |ψiAB = d X j=1 √ pj|ψjiA⊗ |ejiB. (1.24)
It is straightforward to show that this state is normalized, so it is a proper global state vector, and satisfies equation (1.23).
1.1.6 Generalized measurements
The projective measurements are inadequate for describing real experimental measurement for a number of practical reasons. The most fundamental one is that the system of interest is never measured directly, but it interacts with its environment and the experimenter usually observes the effects of the system on the environment. A typical example is an atom that interacts with the continuum of electromagnetic modes and then the radiated field is observed.
Of course this description has only shifts the problem of when to ap-ply the projection postulate, but the point is that the measurement of the system performed in this way can not be described by the formalism of pro-jective measurement. In the following we will develop the theory for such a measurement.
Consider a system S interacting with the environment E (sometimes also called apparatus or meter ) and suppose that at t0 = 0 they are in the
uncorrelated pure state |ΨiSE= |ψiS|θiE. Let them evolve through a global unitary evolution ˆUSE for a certain time t:
|Ψ(t)iSE = ˆUSE(t) |ψiS|θiE. (1.25)
The evolution correlates the system and the environment, then let the en-vironment be measure projectively with the observable ˆR=P
rrπˆr. For
simplicity theπˆrare supposed to be rank-1. Considering the system and the
environment, the whole projector operator is ˆΠr = ˆ1S⊗ ˆπr = ˆ1S⊗ |rihr|E.
After the local measurement on the environment has produced a certain outcome r, the global state is
|Ψ(t)iSEc= ˆ1S⊗ |rihr|E ˆ
USE(t) |ψiS|θiE
pPr(r) , (1.26a)
Pr(r) = SEhΨ(t)| ˆΠr|Ψ(t)iSE. (1.26b)
The global conditional state can actually be rewritten as |Ψ(t)iSEc= 1
pPr(r) Ehr|UˆSE(|ψiS|θiE)
| {z }
= ˆMr|ψiS
|riE (1.27)
where the underlined part is an element of the Hilbert space of the system HS that is linear with respect to the initial state of the system |ψiS and
therefore has been written as ˆMr|ψiS, where ˆMris an operator on HS called
measurement operator.
It is clear that the measurement has disentangled the system and the environment and the probability to obtain the outcome r is
Pr(r) = Ehθ|SψUˆSE 1ˆS⊗ |riEhr|
ˆ
Considering everything depending only on the initial state system |ψiS, the formulae (1.27) and (1.28) become
|ψiSc= Mˆr|ψiS
pPr(r), Pr(r) = Shψ|Mˆ †
rMˆr|ψiS. (1.29)
The generalization to an initial mixed state ρ of the system is
ρc= ˆ Mrρ ˆMr† Pr(r) , Pr(r) = Trh ˆMrρ ˆM † r i . (1.30)
The operators ˆEr= ˆMr†Mˆrare called probability operators (or sometimes
effects for the result r) and the setn ˆEr
o
r is called a positive-operator valued
measurement (POVM). As a matter of fact it is straightforward to show that:
Properties 5. The set of probability operators satisfy the following proper-ties: ˆ Er≥ 0, ∀r; (1.31a) X r ˆ Er= ˆ1S. (1.31b)
These last two properties can also be used to define the POVM thanks to the following:
Theorem 6 (Neumark’s extension). Given a quantum system and a set of positively defined operatorsn ˆFr
o
rthat satisfy the completeness relation (1.31b),
there exists some ancillary system (environment) with Hilber space HE, a
density operator of the environment σ, a global unitary evolution and a pro-jective measurement on the global system (with projectors ˆPr) that produce
exactly the probability operators ˆFr.
The POVMs can be combined together linearly, that is
Properties 7. The set of POVMs is convex, meaning that given two POVMs n ˆEro
r and
n ˆFro
r, then ˆGr≡ p ˆEr+(1−p) ˆFrwith p ∈[0, 1] is still a POVM.
1.2
Bosonic systems
The principal objects of study of this thesis will be bosonic systems and Gaussian states. This section contains the notation used in the rest of it and recalls some fundamental facts.
A system of n bosons can be described by a set of n harmonic oscillators with the non-interacting Hamiltonian (~ = 1):
ˆ H= 1 2 n X k=1 ω qˆ2k+ ˆp2k . (1.32)
The total Hilbert space of this system is H = ⊗n
k=1Fk, where Fk is the
infinite-dimensional Fock space spanned by {|mik}m∈N, that is the eigen-states of the number operator aˆ†kˆak. Canonical position and momentum
operator will be used. ˆ qk= ˆ a+ ˆa† √ 2 , pˆk= ˆ a −ˆa† i√2 , (1.33) [ˆak,ˆa†l] = δkl, [ˆqk,pˆl] = iδkl. (1.34)
The previous formulae can be rewritten defining the vector operator ˆR. ˆ R ≡(ˆq1,pˆ1, . . . ,qˆn,pˆn)T, (1.35) [ ˆRk, ˆRl] = iΩkl, Ω2n≡ n O k=1 0 1 −1 0 , (1.36)
where Ω is the symplectic form of size2n.
The covariance matrix (CM) for the system is
σjk ≡ [σ]jk = h{ ˆRj, ˆRk}i − 2 h ˆRji h ˆRki , (1.37)
where { ˆA, ˆB}= ˆA ˆB+ ˆB ˆA is the anticommutator and h ˆAi ≡Trhρ ˆA i
is the average of the operator ˆA when the system is in the state described by a certain density operator ρ.
Note: with these conventions the CM for a n mode vacuum state is σ=
12n.
There is a stronger form of the Schrödinger uncertainty principle that is called Robertson-Schrödinger uncertainty relation and is a mathematical theorem valid for two generic operators ˆA and ˆB, it states:
4∆2Aˆ∆2B ≥ˆ Dn ˆA, ˆB oE − 2 h ˆAi h ˆBi 2 + Dh ˆA, ˆB iE 2 . (1.38)
It is possible to show (see Ref. [Gos10]) that applying equation (1.38) to the elements of ˆR is equivalent to the more compact form:
σ+ iΩ ≥ 0. (1.39)
From this last form of the Robertson-Schrödinger uncertainty relation it is possible to derive that the CM is always positive defined which is a weaker condition on the CM in classical systems.
Coherent states
Considering for simplicity a single mode (n= 1), the displacement operator can be defined as
ˆ
D(α) = exphαˆa†− α∗ˆa i
, α ∈ C. (1.40)
Properties 8. Some important properties of ˆD(α) are: (a) ˆD(α) is unitary;
(b) ˆD†(α) = ˆD−1(−α); (c) ˆD†(α)ˆa ˆD(α) = ˆa+ α.
The coherent states are defined as the states obtained applying a dis-placement operator on the vacuum state.
|αi ≡ ˆD(α) |0i . (1.41)
Properties 9. Some important properties of the coherent states are: (a) |αi are eigenvalues of ˆa, that is ˆa |αi= α |αi;
(b) expanding a coherent state with the orthonormal base of {|ki}k∈N gives
|αi = e−|α|22 ∞ X k=0 αk √ k!|ki ; (1.42)
(c) {|αi}α∈C form an overcomplete basis, that is they are not orthogonal:
hα|βi = e−12(|α|
2+|β|2+α∗β)
, but any state can be expanded using the set of the coherent states: π1 R |αihα| d2α= 1;
(d) they are minimum uncertainty states: σ|αi= 1 = σvacuum and
uncer-tainties for position and momentum are equal. Thermal states
The thermal state for a single mode at temperature T = 1/(KBβ) is
ρth= e−βa†a Tre−βa†a = ∞ X m=0 Nm (N + 1)m+1 |mihm| , (1.43)
where N = (eβ−1)−1is the average number of quanta at thermal equilibrium
and the expansion with the Fock states is shown. The covariance matrix can be easily found to be
σth=2N + 1
0
0 2N + 1
1.2.1 Characteristic function and Wigner function
The characteristic function of a certain density matrix is defined to be:
χρ(α) ≡ Tr[ ˆD(α)ρ], (1.45)
while the Wigner function is defined as its Fourier transform: Wρ(β) ≡
1 π
Z
d2α χ(α) eβ∗α−βα∗. (1.46) The Wigner function is a normalized quasi-probability distribution for the quantum state ρ in the phase space defined by(x = Re [β] , p = Im [β]). This is particularly convenient because the marginals of the Wigner function gives the probability distribution of position and momentum, that is:
Z
Wρ(x, p) dx = hp|ˆρ|pi
Z
Wρ(x, p) dp = hx|ˆρ|xi (1.47)
The reason why the Wigner function is not a proper probability distribution is that there exists some states for which Wρ(β) takes negative values.
General case for n modes
In a general state of n bosons the total displacement operator can be writ-ten as ˆD(α) = ⊗n
k=1Dˆk(αk), this means that the k-mode is displaced with
parameter αk∈ C.
The definitions of characteristic function and Wigner function still ap-plies. χρ(α) = Tr h ρ ˆD(α)i, (1.48) Wρ(β) = Z d2nα π2n exp h α†β+ β†αiχρ(α). (1.49)
In particular the Wigner function will depend on n complex parameters (β1, . . . , βn); these can be written in Cartesian form βk= (xk+ iyk)/
√ 2 so that Wρ(β) depends equivalently on the column vector X = (x1, y1, . . . , xn, yn)T.
On the other hand the characteristic function can be considered depending on n complex parameters αk or, in Cartesian form αk = (ak+ ibk)/
√ 2, on the column vector Λ= (a1, b1, . . . , an, bn)T.
The set of displacement operator is said to be complete, meaning that any operator ˆO on the total Hilbert space H can be written as
ˆ O = Z d2nα πn Trh ˆO ˆD(α)i ˆD † (α) = Z d2nα πn χOˆ(α) ˆD † (α). (1.50)
Equation (1.50), often referred to as Glabuer formula, shows how any opera-tor is completely described by its characteristic function χOˆ(α) (or alterna-tively, by its Fourier transform, i.e. the Wigner function).
Trace rule in the phase space
The characteristic function and the Wigner function allow to evaluate op-erators’ traces as integrals in the phase space, this is particularly useful to calculate correlation functions and statistics of a measurements when Gaus-sian states (see section 1.2.4 on page 17) and operators are involved. The proof of the following formulae can be found in Ref. [FOP05].
Trh ˆO1Oˆ2 i = Z d2nα πn χOˆ1(α) χOˆ2(−α) = = πn Z d2nβ WOˆ1(β) WOˆ2(β). (1.51) 1.2.2 Symplectic transformation
Symplectic transformation arises very naturally in classical mechanics. Given a system with canonical coordinates(q1, . . . , qn), conjugate momenta (p1, . . . , pn)
and Hamiltonian H, its evolution is described by Hamilton’s equations: ˙pk = − ∂H ∂˙qk , ˙qk= ∂H ∂˙pk . (1.52)
If one defines the vector R= (q1, p1, . . . , qn, pn)T, the previous equations can
be written in a more compact form ˙
Rk= Ωkl
∂H ∂Rl
, (1.53)
where Ω is defined in equation (1.35).
Let us suppose that a linear change of coordinate is made with the matrix F , and we can impose that the new evolution equations keeps the same form.
R0 = F R, Fkl=
∂Rk0 ∂Rl
. (1.54)
The time evolution of the new coordinates is: ∂R0k ∂t = ∂R0k ∂Rl ∂Rl ∂t = = FklΩlm ∂H ∂Rm = = FklΩlm ∂R0n ∂Rm ∂H ∂R0n = = FklΩlmFnm ∂H ∂R0 n . (1.55)
If we want the new coordinate to obey to Hamilton’s equation, it must be FklΩlmFnm = Ωknor, using the matrices:
F ΩFT= Ω. (1.56)
It can be shown quite straightforwardly that the set of F that satisfy condition (1.56) form a group with the usual matrices product, and is called symplectic group,Sp(n, R) with dimension (n + 1)n/2, where n is the dimen-sion of the real square matrices studied so far.
The symplectic group can be extended considering also the translation of the R vectors: R 7→ R+ d, where d is a real vector. This gives the affine symplectic group ISp(n, R).
1.2.3 Linear and bilinear interaction of modes
The interaction Hamiltonians that will be considered are at most bilinear in the modes, this means it can be written in the most general form:
ˆ H = n X k=1 gk(1)ˆa†k+ n X k>l=1 g(2)kl ˆa†kˆal+ n X k,l=1 gkl(3)ˆa†kˆa†l + h.c. (1.57)
When considering the unitary transformation associated with this Hamil-tonian, the three parts can be separately analysed:
• the first block g(1)ˆa†+ h.c. corresponds to the displacement operators
already defined;
• the second block contains terms of the kind ˆa†a that are those of freeˆ evolution, while the ones ∝ˆa†ˆb + h.c. describe the linear mode mixing performed through a beam splitter ; this interaction is called passive as it conserves the total number of quanta;
• the third block has terms ∝ g(3)(ˆa†)2+h.c which generates single-mode
squeezing and also ∝ g(3)ˆa†ˆb†+h.c. that generates two-modes squeezing. These three kinds of unitary evolution together belong to the unitary representation of the affine symplectic group ISp(2n, R), which is called the metaplectic representation. A generic element of ISp(2n, R) transforms the phase space in the following way:
X 7→ F X+ d (1.58)
where F is an element ofSp(2n, R) =F ∈ GL(2n)|F Ω2nFT = Ω2n and
d is a real column vector with 2n elements. Covariance matrix evolve ac-cordingly:
Beam splitter
The simplest two-mode interaction is the linear mixing described by H ∝ ˆ
a†ˆb + ˆb†ˆa. For a radiation field it corresponds to the linear interaction tak-ing place in a linear optical medium, that is a beam splitter. The unitary evolution can be written as
ˆ
U(ζ) = exphζaˆ†ˆb − ζ∗ˆaˆb† i
, (1.60)
where ζ = φeiθ is the coupling constant, proportional to the interaction time
and the linear susceptibility of the medium.
Using the Schwinger two mode boson representation of SU(2) algebra to make the identifications: J+ = ˆa†ˆb, J−= ˆaˆb†, J3 = 21[J+, J−] = 12(ˆa†ˆa − ˆb†ˆb),
it is possible to disentangle the evolution operator into ˆ
U(ζ) = exphζaˆ†ˆb − ζ∗aˆˆb†i= exp [ζJ+− ζ∗J−] =
= expheiθtan φ ˆa†ˆb i
(cos2φ)ˆb†ˆb−ˆa†ˆaexph−e−iθtan φ ˆaˆb† i
= = exph−e−iθtan φ ˆaˆb†i(cos2φ )ˆa†ˆa−ˆb†ˆbexpheiθtan φ ˆa†ˆbi.
(1.61)
The constant τ ≡cos2φ is called transmissivity of the beam splitter.
Using the Baker–Campbell–Hausdorff formula: eα ˆABeˆ −α ˆA= ˆB+ αh ˆA, ˆBi+ α 2 2! h ˆA,h ˆA, ˆB ii +α 3 3! h ˆA,h ˆA,h ˆA, ˆB iii + . . . , (1.62) the Heisenberg evolution of modesˆa and ˆb is given by
ˆ U†(ζ)ˆa ˆb ˆ U(ζ) = Sζ ˆa ˆb (1.63) where Sζ=
cos φ eiθsin φ −eiθsin φ cos φ
. (1.64)
The corresponding evolution for the position and momentum is given by the4 × 4 symplectic matrix
Fζ=
Re [Sζ] − Re [Sζ]
Im [Sζ] Im [Sζ]
. (1.65)
If ζ is real then the symplectic matrix can be written as a function of the transmissivity FU (τ )ˆ = √ τ 0 √1 − τ 0 0 √τ 0 √1 − τ −√1 − τ 0 √τ 0 0 √1 − τ 0 √τ . (1.66)
One-mode squeezing
The Hamiltonian for this transformation is H ∝ g(3)(ˆa†)2+h.c, so the unitary
evolution is S(ξ) = exp 1 2 ξ(ˆa†)2− ξ∗ˆa2 . (1.67)
In general ξ ∈ C, so we can write it as ξ = reiφ, then the evolution of the
mode operators turn out to be
S†(ξ)ˆaS(ξ) = cosh rˆa+ eiφsinh rˆa†, (1.68)
S†(ξ)ˆa†S(ξ) = cosh rˆa†+ e−iφsinh rˆa, (1.69) or in matrix form S†(ξ) ˆa ˆ a† S(ξ) = S ˆa ˆ a† (1.70) where the matrix S is defined to be
S =
cosh r eiφsinh r e−iφsinh r cosh r
. (1.71)
The symplectic matrix that transforms the position and momentum is FS(ξ) =
cosh r + Reeiφsinh r
Imeiφsinh r Imeiφsinh r
cosh r − Reeiφsinh r . (1.72) If φ= 0 then FS(r)= er 0 0 e−r , (1.73)
and the corresponding Hamiltonian is H= ir
2t((ˆa
†)2− ˆa2). (1.74)
The transformation of the variance of position and momenta can be calcu-lated with σ0 = F σFT one obtains
(∆q)27→ e2r(∆q)2, (∆p)27→ e−2r(∆p)2. (1.75) So in the case of r > 0 the effect of the squeezing term is to reduce the variance of the momentum, and obviously to increase the variance of the position, accordingly to Heisenberg’s uncertainty principle.
In a more general case it is possible to consider the quadrature operator ˆ xθ = ˆ aeiθ+ ˆa†e−iθ √ 2 . (1.76)
It is clear thatxˆθ=0= ˆq and xˆθ=π/2= ˆp and for a generic θ it is some linear
To see what happens when ξ is not real, let us define the squeezed vacuum state as the state obtained applying the squeezing operator on the vacuum state: |ξi = S(ξ) |0i = √ 1 cosh r ∞ X k=0 eiφsinh r 2 cosh r kp(2k)! k! |2ki , (1.77)
where the expansion is in the number basis. The mean photon number is
hξ|ˆa†ˆa|ξi= | sinh r|2, (1.78) in fact this interaction does not conserve the number of photons and to squeeze the state, energy must be put into the system.
It is straightforward to show that
hξ|ˆxθ|ξi = 0, (1.79)
(∆ˆxθ)2 = hξ|ˆx2θ|ξi =
1 2 e
2rcos2(θ − φ/2) + e−2rsin2(θ − φ/2) . (1.80)
So squeezed vacuum is a minimum uncertainty state for the pair of ob-servablesxˆφ/2 and xˆφ/2+π/2, since
(∆ˆxφ/2)2= 1 2e 2r, (∆ˆx φ/2+π/2)2 = 1 2e −2r, (1.81) and the variance for xˆφ/2+π/2 decreases under the variance for the simple
vacuum state, which explains the term "squeezed". Squeezing is a non-classical property of continuous-variable quantum states and, apart from being and interesting fundamental feature of quantum mechanics, it is also relevant for different applications in quantum technologies. In particular squeezed states can be used to generate continuous-variable entanglement and they are widely used for quantum metrological applications (Ref. [Cav81] and [AAA13]).
Two-mode squeezing
The last unitary evolution is the one that corresponds to the Hamiltonian ∝ ˆa†ˆb†+ h.c.:
S2(ξ) = exp
h
ξˆa†ˆb†− ξ∗ˆaˆbi, (1.82) where ξ = re1φ as is the previous section. It is possible to show that the
Heisenberg evolution of the field modes is S2†(ξ) ˆaˆb† S2(ξ) = S ˆa ˆb† , (1.83)
where
S =
cosh r eiφsinh r e−iφsinh r cosh r
. (1.84)
The number of quanta is not conserved, but the difference in mean photon number si conserved.
The corresponding symplectic matrix is FS2(ξ) = cosh r12 T2 T2 cosh r12 , (1.85) where T2 =Re eiφsinh r Imeiφsinh r Imeiφsinh r Reeiφsinh r . (1.86)
1.2.4 Definition of Gaussian states
A state is called Gaussian state if its Wigner function is a Gaussian, that is:
Wρ(X) = 2 π nexp h −12 X − ¯XT σ−1 X − ¯Xi pDet[σ] (1.87)
and as such, it is uniquely defined by the quadratures’ averages ¯X and the σ.
It is worth noting that the only pure states for which the Wigner function is always positive are the Gaussian states (see Ref. [LB95]). The positivity allows one to interpret the quasi-probability distribution as a proper prob-ability distribution for the state averages hˆqi and hpi and think the phaseˆ space as an analogue for the classical one.
Proposition 10. The previously introduced coherent states and thermal states are Gaussian states.
Evolution of Gaussian states In general, under a symplectic transforma-tion in the phase space the characteristic functransforma-tion and the Wigner functransforma-tion transform as scalars, namely
χU ρ ˆˆ U†(Λ) = χρ(F−1Λ), (1.88) WU ρ ˆˆ U†(X) = χρ(F−1X), (1.89) in fact χU ρ ˆˆ U†(Λ) = Trh ˆU ρ ˆU†D(Λ)ˆ i = Trhρ ˆU†D(Λ) ˆˆ Ui= Trhρ ˆD(F−1Λ)i. (1.90) For Gaussian states this transformation changes the first and second mo-ments as stated in equations (1.58) and (1.59).
Theorem 11 (Williamson). Given a real, symmetric and positive defined 2n × 2n matrix V , there exist an F ∈ Sp(2n, R) and a 2n × 2n matrix
W = n L j=1 dj1 0 0 1
with dj ∈ R called symplectic eigenvalues such that
V = FTW F . (1.91)
F and {dj}j=1,...,n are unique up to a permutation of the elements of the dj.
See Ref. [Wil].
Proposition 12. The symplectic eigenvalues have the following properties: • the dj are the eigenvalues of |iΩV |;
• the Robertson-Schrödinger uncertainty relation is equivalent to saying that the symplectic eigenvalues are ≥1.
The matrix W is the CM of a thermal state with n modes, each with temperature βj such that dj = 2Nj+ 1 with Nj = (eβj− 1)−1.
Williamson’s theorem ensures that every Gaussian state centered in(0, 0) (then uniquely defined by its σ) can be obtained starting form a thermal state and applying some unitary evolution corresponding to an Hamiltonians of the form (1.57). The resulting state then can be translated wherever in the phase space through the appropriate displacement operator.
So a generic Gaussian ρG can be written as
ρG= ˆUSρthUˆS†, UˆS = exp [−iHbil] , (1.92)
for a suitable choice of ρth (in the form of equation (1.43)) and Hbil (in the form of equation (1.57)).
From an applicative point of view this is extremely important as it means that any Gaussian state can be generated only using linear and bilinear interactions.
Single-mode Gaussian state
Using Williamson’s theorem ( 11) the decomposition for a single-mode Gaus-sian state is
ρG = ˆD(α) ˆS(ξ)ρth(¯n) ˆS†(ξ) ˆD†(α), (1.93)
which is called displaced squeezed thermal state (DSTS). From this formula it is clear that the Gaussian state is uniquely defined by the average pho-ton number of the thermal staten, and by the displacement and squeezing¯ parameters α and ξ.
Starting from the thermal state’s covariance matrix (equation (1.44)), the CM of a DSTS is given by equation (1.72): σ = FS(ξ)σthFS(ξ) and
gives
σ = (2¯n+ 1)cosh(2r) + sinh(2r) cos φ − sinh(2r) sin φ − sinh(2r) sin φ cosh(2r) − sinh(2r) cos φ
. (1.94) The average photon number is given by
TrhρGˆa†ˆa i = (¯n+1 2) cosh(2r) − 1 2+ |α| 2, (1.95)
while the purity is
µ(ρG) =
1
2¯n+ 1; (1.96)
this means that the purity depends only on the thermal state and not on the displacement or the squeezing. This makes sense, since these transformations do not change the trace involved in the definition of purity.
Two-mode Gaussian states
This section studies a bipartite Gaussian state with n+ m modes, the global Hilbert space is HA⊗ HB. Given two global density operators ρ1, ρ2, they
are said to be locally equivalent if there exist two unitary transformations ˆ
UA acting on HA and ˆUB acting on HB such that
ρ2 = ˆUa⊗ ˆUB ρ1 UˆA† ⊗ ˆUB†. (1.97)
Since the tensor product of the Hilbert spaces corresponds to a direct sum of the phase spaces, this means that the covariance matrix σ1, σ2 of
respectively ρ1 and ρ2, are related by
σ2 = Slocσ1Sloc, (1.98)
where Sloc= SA⊗ SB is an element of the subgroup Sp(2n, R) ⊗ Sp(2m, R)
and SA, SB are the symplectic matrix corresponding to the unitary local
evolutions ˆUA, ˆUB respectively.
A generic CV of a m+n modes Gaussian state is a 2m+2n square matrix
σ=A C
C B
, (1.99)
where A and B are the CM of the reduced systems states A and B, and they are square matrices with respective dimensions2n and 2m, C represents the correlations between the two part of the system and is a2n × 2m matrix.
There are four local symplectic invariants under operations belonging to Sp(2n, R) ⊗ Sp(2m, R):
In the case where n= m = 1, that is a global two-mode system, it can be shown (see Ref. [Dua+00]) that any CM can be transformed in the canonical form σcan= a 0 c1 0 0 a 0 c2 c1 0 b 0 0 c2 0 b . (1.101)
The four parameters are related to the four local symplectic invariants: I1= a2, I2 = b2, I3= c1c2, I4= (ab − c21)(ab − c22), (1.102)
and the uncertainty principle can be rewritten as
I1+ I2+ 2I3≤ I4+ 1. (1.103)
Purity of Gaussian states
The purity of a quantum system is defined to be
µ= Trρ2 . (1.104)
The main property of µ is that it is always ≤1 and the equality holds iff the system ρ is a pure state, hence the use of it as a marker for pure states. For Gaussian states it can be written using the formula:
Tr [ρ1ρ2] = πn
Z
d2nβWρ1(β)Wρ2(β), (1.105)
and reduces simply to
µ= 1
pDet [σ]. (1.106)
It is clear that purity is preserved by unitary transformations. Consid-ering that the vacuum state is the only pure thermal state then all pure Gaussian states are obtained as
|ψGi = ˆUS|0i . (1.107)
Quantifying squeezing of Gaussian states (decibel) Considering for simplicity a single-mode Gaussian state, its CM gives all the information about the fluctuations of every quadrature in the phase space and is a quadratic form in hˆqi and hˆpi. Diagonalizing the CM, one obtains the two ordered eigenvalues λ1, λ2 and their relative eigenvectors in the phase space
x1, x2, that define a new orthogonal basis. The Gaussian in the phase space
in this new basis is the product of two one-dimensional Gaussians. This means that the vector x1 gives the quadrature with the minimum variance.
From this it follows that to understand if a Gaussian states has some squeezed quadrature (that is, with variance less that the variance of the vacuum state) it is enough to study the minimum eigenvalue of the CM.
The squeezing of a state is often expressed in decibel, comparing it with the variance of the vacuum state, defining the quantity
SqdB(ρ) ≡ −10 log10
λ1
λvacuum
Open quantum systems and
continuous-measurement
In this chapter we will introduce some concepts that will be useful to obtain our results. First we will describe in section 2.1 the general form of a Marko-vian master equation describing an open quantum system, providing then a relevant example in section 2.1.1 (a bosonic mode interacting with a finite temperature environment). Then, in section 2.2 we will introduce two exam-ples of instantaneous quantum measurement for continuous-variable systems (photodetection and homodyne detection). Finally, we will consider the evo-lution of a quantum system subjected to a continuous quantum measurement. We will first provide two examples: the first will describe the effect on a quan-tum state subjected to a continuous position monitoring, while in the second we will explicitly derive the stochastic master equation for a bosonic mode interacting with an environment which is continuously monitored through a homodyne detection. Both these examples will be generalized in the last section, which will provide the general mathematical formalism describing continuous-measurement and feedback control for linear quantum system.
2.1
Markovian master equation
In the case of a close system with Hamiltonian H the time evolution of a density operator can be derived from Schrödinger’s equation and it is a linear first-order equation:
˙ρ = −i [H, ρ(t)] . (2.1)
On the other hand, when the system S is interacting with an environment E, the linearity of the evolution is lost when calculating the partial trace on the environment’s degree of freedom:
ρS(t) = TrE[|ψ(t)iSEhψ(t)|] . (2.2)
However there is an interesting case where the evolution of an open system remains linear: when the environment is a Markovian bath, that is when E at any time does not have any memory about previous interaction with S. This idea can be expressed more formally and it is possible to derive that the only physical evolutions (namely the quantum channels studied in section 1.1.4 on page 3) that satisfy this condition are (see Ref. [Lin76])
˙ρ = Lρ = −ih ˆH, ρi+X
k
D[ˆck]ρ. (2.3)
L is a superoperator called linbladian and the dissipative part that makes the evolution non unitary are superoperators of the form
D[ˆck]ρ = ˆckρˆc † k− 1 2 n ˆ c†kˆck, ρ o . (2.4)
Equation (2.3) is called Markovian master equation. For the derivation, see appendix A on page 62.
It is worth noting that the above representation of the evolution super-operator L is not unique and we can reduce the ambiguity by requiring that the operatorsˆck are linearly independent, but the representation is still not
unique, as it remains the same under unitary transformations of theˆck, that
is
ˆ
ck→ Tklcˆl, (2.5)
where Tkl is a unitary matrix, and also for c-number shifts of the ˆck
accom-panied by a new term of the Hamiltonian, namely ˆ ck→ ˆck+ χk, (2.6a) ˆ H → ˆH+ i 2(χ ∗ kˆck+ h.c.). (2.6b)
2.1.1 An example: the bosonic lossy channel
In this section we want to derive equation (2.3) in the special case of bosonic systems. Consider a system S consisting of a single harmonic oscillator inter-acting with a heat bath B at temperature T that is made of a large number of harmonic oscillators, as, for example, the modes of the free electromag-netic field or phonon modes in a solid. The free Hamiltonian for each system is then considered to be:
HS = ω0 ˆ a†ˆa+1 2 , HB= X j ωj ˆb† jˆb + 1 2 , (2.7)
whereˆa, ˆa† = 1 andhˆb j, ˆb†k
i
Let us consider the coupling part of the total Hamiltonian to be V =X j gj ˆ a†ˆbj+ ˆaˆb†j . (2.8)
Passing to the interaction picture respect to the free part of the Hamiltonian H0 = HS+ HB, the interaction becomes
V = ˆa†Γ(t)eiω0t+ ˆa†Γ†(t)e−iω0t, whereΓ(t) =X
j
gjˆbje−iωjt. (2.9)
As the interaction Hamiltonian is quadratic, we know that the quantum map describing the evolution will be Gaussian, that is it will send initial Gaussian states into Gaussian states.
Let us now assume that the coupling is weak and that short-time cor-relation between the system and the bath are negligible (that is, Markov approximation), one can obtain the evolution of the global system. Then, by tracing out the degrees of freedom of the bath, we obtain the evolution of the system to be
˙ρ = γ(N + 1)D [ˆa] ρ + γN Dhˆa† i
ρ, (2.10)
where the operator D is the same defined in equation (2.4).
It is straightforward to see that the rate of change of the average photon number hniˆ = hˆa†ˆai in the cavity is given by
dhˆa†ˆai dt = Tr ˆ a†ˆadρ dt = −γ hˆa†ˆai+ γN, (2.11) whose solution is hn(t)i = hn(0)i e−γt+ N (1 − e−γt). (2.12) The evolution of the CM can be calculated in the same way to be:
σ(t) = e−γtσ(0) + (1 − e−γt)σ(∞). (2.13) For t → ∞, no matter what the initial CM is, the state will asymptoti-cally tend to σ(∞), that is the CM of a thermal state of the same temper-ature as the bath. This CM has already been studied and it is 12(2N + 1), with mean number of quanta N .
Zero temperature In the special case of T = 0, also N = 0, the master equation becomes
ρ |0i BS η ˆb ˆ a
Figure 2.1: Model for a loss master equation at zero temperature.
This master equation is called loss master equation and describes optical communication through a lossy fiber, since thermal photons are negligible at room temperature.
The analytical solution of the zero temperature master equation can be found to be ρ(t) = ∞ X n=0 ˆ Vnρ(0) ˆVn†, where ˆVn= r η−1− 1 n! ˆa nηˆa†ˆa/2, (2.15)
where the constant η= e−γt.
This quantum channel can also be modelled through the action of an unbalanced beam splitter with transmissivity η that mixes the density op-erator ρ of the modeˆa and a second single-mode ˆb in the pure state |0ibh0|, and then tracing out the ˆb degree of freedom (see Ref. [Gen]). The scheme is showed in figure 2.1. ρ(η) = Trbh ˆU(φ)(ρ ⊗ |0ih0|) ˆU†(φ) i = = ∞ X n=0 tan2nφ p! ˆa n(cos φ)aˆ†ˆa ρ(cos φ)ˆa†ˆaˆa†n= = ∞ X n=0 ˆ Vnρ ˆVn†, (2.16)
where η= cos2φ,(0 ≥ η ≥ 1) and operator ˆU(φ) is the operator defined in
equation (1.61) for the beam splitter interaction in the bilinear Hamiltonian. It can be shown that, if the mode ˆb is in the thermal state with average quanta number N , the same scheme describes the general case of non-zero temperature and thermal noise.
2.2
Quantum measurement of continuous-variable
systems
In this section we will describe two examples of quantum measurement for bosonic quantum systems: photodetection and homodyne detection.
2.2.1 Photodetection
Photodetection is a very important example of a non-Gaussian quantum measurement. It is essentially the projective measurement associated with the number state basis, so the set of projectors is {|nihn|}n. Experimentally photons are revealed thanks to their interactions with atoms or molecules. When a photon is energetic enough to ionize a single atom or promotes an electron to a conduction band, the electron can start an avalanche process that creates enough charge to be detectable. Every realistic photodetector has a non-unit efficiency η that must be taken into account when modelling the measurement, that is only a fraction η of incoming photons are actually detected.
Studying the process of the excitation of an electric field, it is possible to calculate the probability distribution of the photon count and derive for an inefficient photodetector the following probability operators:
ˆ Πm(η) = ηm ∞ X k=m (1 − η)k−m k m |kihk| . (2.17)
It is possible to show that this is a proper POVM, that is, it satisfies the properties of equations (1.31), but the inefficiency causes them not to be orthogonal, while the POVM of an ideal detector with unity efficiency is
ˆ
Πm(η = 1) = |mihm| . (2.18)
The characteristic and the Wigner function of the operator in equation (2.18) can be calculated to be χ|mihm|(α) = hm| ˆD(α)|mi = e− 1 2|α| 2 Lm(|α|2), (2.19a) W|mihm|(β) = 2 π hm|(−) ˆ a†ˆaD(2β)|mi =ˆ 2 π(−) ke−2|β|2 Lm(4|β|2), (2.19b)
where Lm(x) are the Laguerre polynomials. From these it is possible to
obtain the characteristic and Wigner function of the POVM for the inefficient photodetection (equation (2.17)): χΠˆm(η)(α) = 1 ηLm |α|2 η exp −2 − η 2η |α| 2 , (2.20a) WΠˆm(η)(β) = 2 π (−)mηm (2 − η)m+ 1Lm 4|β|2 2 − η exp −2 − η 2η |β| 2 . (2.20b)
An inefficient photodetector can be also described by an ideal photode-tector that receives the signal through a beam splitter with transmissivity equal to the quantum efficiency η and receiving the signal and an auxiliary mode in the vacuum state |0ih0|, that is, the scheme described in section 2.1.1 on page 24.
Beside inefficiency, imperfect photodetector have also an other problem: dark counts, i.e. signals form the photodetector which do not correspond to any actual incoming photon. This can be described by the same scheme de-scribed in the previous paragraph, but with the auxiliary mode in a thermal state or a phase-averaged coherent state, respectively describing thermal and Poissonian background noise, as analysed in Ref. [FOP05].
2.2.2 Homodyne detection
Homodyne detection scheme are used to provide the measurement of a single-mode quadraturexˆφ (defined in equation (1.76)). To achieve this goal, the
signal is mixed with a high-intensity classical field at the same frequency, referred to as the local oscillator (LO). Homodyne detection was proposed for the radiation field in Ref. [YC83], and demonstrated in Ref. [ACY83].
A homodyne detection is called balanced homodyne detection when the mixing is obtained through a 50/50 beam splitter, as in figure 2.2 on the next page. The mode ˆb is the LO of the detector, it has the same frequency as modeˆa and it is in a coherent state |zi with large amplitude z. The BS has a real coupling, so it does not add phase-shifting. Since in experiments the two modes are usually generated by the same source, we will assume that they have a fixed phase relation and use the LO as a reference for the quadrature measurement.
In the following we will show that tuning the relative phase φ = arg[z], it is possible to measure different quadratures xˆφ of the mode ˆa. After the
BS the two modes c and ˆˆ d are detected by two identical photodetectors and finally the difference of photocurrents at zero frequency is electronically processed and rescaled by2|z|.
The action of the BS on the two incoming modes is known from equa-tion (1.63), from this it is possible to calculate the resulting homodyne pho-tocurrent I to be ˆ I = ˆc †c − ˆˆ d†dˆ 2|z| = ˆ a†ˆb + ˆb†ˆa 2|z| . (2.21)
The spectrum of ˆa†ˆb + ˆb†ˆa is the set Z of relative integers, therefore the spectrum of ˆI is discrete too, but in the limit of a highly excited LO, i.e. z 1, it tends to the real axis. Since the mode ˆb is in a strong semi-classical state, we can neglect the quantum fluctuations of the LO and substitute ˆb → z, ˆb† → z∗. In this case, it is straightforward to see that the moments
|zi BS50/50 ˆb ˆ d ˆ a ˆ c I
Figure 2.2: Scheme for homodyne detection
of the homodyne photocurrent are ˆ I = ˆxφ, Iˆ2 = ˆx2φ+ ˆ a†ˆa 4|z|2, . . . , Iˆ n= ˆxn−2 φ ˆ x2φ+ ˆa †ˆa 4|z|2 . (2.22) We can assume that hˆa†ˆai 4|z|2, then the distribution of the outcomes
I of the homodyne photocurrent is equal to that of the corresponding field quadrature. The POVM { ˆΠI} of the detector coincides with the spectral
measure of the quadratures: ˆ
ΠI → ˆΠ(x) = |xiφhx| = δ(ˆxφ− x), (2.23)
i.e. the projector on the eigenstate of the quadraturexˆφwith eigenvalue x.
This last equation means that, in the strong LO regime, the balanced homodyne detector achieves the ideal measurement of the quadrature xˆφ.
It must be noted that, in order for this to be true, the amplitude of the coherent state must be both much bigger than1 and much bigger than the average number of photons hˆa†ˆai.
Inefficient homodyne detection We describe an inefficient detector as an ideal detector preceded by a BS with transmissivity η and the second port left in the vacuum. Then the difference of photocurrents must be rescaled by a factor2|z|η. In this case the POVM operators given in equation (2.23)
become: ˆ Πη(x) = 1 q 2πδ2 η exp −(ˆxφ− x) 2 2δ2 η = = q1 2πδ2 η Z C dy exp −(y − x) 2 2δ2 η |yiφhy| , (2.24) where δη2= 1 − η 4η . (2.25)
In other words, the POVM operators in the inefficient case are the Gaussian convolution of the ideal ones.
The Wigner function of the inefficient homodyne POVM can be calcu-lated to be WΠˆη(x)(β) = 1 q 2πδ2 η exp −(x − (βe −iφ+ β∗eiφ)/2)2 22δ2 η . (2.26)
For the efficient detection: WΠˆη=1(β) → δ x −1 2(βe −iφ + β∗eiφ) . (2.27)
2.3
Continuous measurement and stochastic
mas-ter equations
A continuous measurement is one in which the information extracted from the system goes to zero as the duration of the measurement goes to zero. As discussed in section 1.1.5 on page 5, the measurement changes the knowledge about the system, but the measurements considered so far are somehow instantaneous, i.e. the duration of the measuring process is much faster than the typical times of evolution of the system. In this section we will study the evolution of a system subjected to a measurement process that is not instantaneous and last a finite time length.
We will firstly present two examples, the first being the continuous mon-itoring of the position observable and then we will study a continuous homo-dyne detection. In both cases we will derive the stochastic master equation (SME) that describes the evolution of the density operator of the system. Eventually we will present the general theory of a system subjected to con-tinuous monitoring and the most general form of SME.
2.3.1 SME for continuous position monitoring The derivation presented in this section follows Ref. [JS06].
Let ˆX be an observable on the system, with a continuous spectrum and eigenvectors {|xi} that satisfy hx|x0i = δ(x−x0). The measurement operators
associated with it are ˆMx = |xihx|. The time in which the measurement
happens is divided into equal intervals of length∆t. In each interval we will apply the measurement operator:
ˆ A(α) =r 4k∆t4 π Z +∞ −∞ exp−2k∆t(x − α)2 |xihx| d x, (2.28)
which is a Gaussian weighted sum of projectors onto the eigenstates of ˆX with mean α and variance(4k∆t)−1, the reason of this latter choice will be clear when the limit∆t → dt will be taken. This is a new set of POVM, and the possible outcomes are α ∈ R. If the system is in the state |ψi=R ψ(x) |xi dx, the probability distribution of the new POVM is
Pr(α) = Trh ˆA†(α) ˆA(α) |ψihψ|i= =r 4k∆t π Z +∞ −∞ |ψ(x)|2exp−4k∆t(x − α)2 dx, (2.29)
and the average value of α is hαi = Z +∞ −∞ αPr(α) dα = =r 4k∆t π Z +∞ −∞ Z +∞ −∞ α|ψ(x)|2exp−4k∆t(x − α)2 dx dα = = Z +∞ −∞ x|ψ(x)|2dx = h ˆXi . (2.30)
Since in the end the limit ∆t → dt will be considered, we can assume that the Gaussian is much broader than the |ψ(x)|2 and approximate the
latter in equation (2.29) as a δ-function centred in h ˆXi.
Pr(α) ≈r 4k∆t π Z +∞ −∞ δ(x − h ˆXi) exp−4k∆t(x − α)2 dx = =r 4k∆t π exp h −4k∆t(α − h ˆXi)2i. (2.31)
It is then possible to consider α as a stochastic quantity distributed along a normal distribution with mean h ˆXi and variance(8k∆t)−1.
It is useful to consider a new stochastic quantity ∆W , derived from α, that is distributed along a Gaussian distribution with mean 0 and variance √
∆t. ∆W is related to α in the following way: α= h ˆXi+√∆W
8k∆t. (2.32)
According to the theory of POVM, after a measurement ˆA(α) is made, the state of the system at time t+ ∆t is (cfr. (1.29)):
|ψ(t + ∆t)i ∝ ˆA(α) |ψ(t)i ∝
∝ exph−2k∆t(α − ˆX)2i|ψ(t)i ∝
∝ exph−2k∆t ˆX2+ ˆX(4k h ˆXi∆t +√2k∆W )i|ψ(t)i ≈
≈h1 − 2k∆t ˆX2+ ˆX4k h ˆXi∆t +√2k∆W + k ˆX(∆W )2i|ψ(t)i , (2.33) where the approximation comes from expanding the exponential in the first order of∆t and second order of ∆W . This choice will be clarified later.
To have a continuous measurement we must take the limit ∆t → dt. In this limit more and more measurement are made in any finite time interval, but the choice of ˆA(α) is made in such a way that each measurement is weaker and weaker, so in the end a sensible result is obtained.
Note: it will be assumes for now that∆W → dW and (∆W )2→ dt. The
proof of this is discussed in properties 13 on page 33 and 14 on page 34. The evolution of the state due to the measurement is:
|ψ(t + dt)i = h
1 − (k ˆX2− 4k ˆX h ˆXi) dt +√2k ˆXdWi|ψ(t)i
phψ(t + dt)|ψ(t + dt)i . (2.34)
The normalization of |ψ(t + dt)i can be expanded to first order in dt and second order indW to finally obtain:
d|ψi ≡ |ψ(t + dt)i−|ψ(t)i =h−k( ˆX − h ˆXi)2dt +√2k( ˆX − h ˆXi) dWi|ψ(t)i . (2.35) This is called the Schrödinger stochastic equation (SSE) and describes the evolution of the state of the system when a continuous measurement is per-formed.
In each interval of timedt we are obtaining a certain measurement result dy(t) = h ˆXidt + √dW
8k, (2.36)
and the quantity y(t) is called measurement record as it is the record of the results obtained up to the time t, but is also sometimes called current because
in the case of quantum optics, it corresponds to a measured photocurrent (cfr. [Wis94]).
The SSE can also be written in terms of the density operator ρ= |ψihψ| and the new equation derived from (2.35) is referred to as stochastic master equation (SME).
dρ ≡ ρ(t + dt) − ρ(t) =
= (d|ψi) hψ| + |ψi (dhψ|) + (d|ψi)(dhψ|) =
= −kh ˆX,h ˆX, ρiidt +√2k( ˆXρ+ ρ ˆX −2 h ˆXi ρ) dW.
(2.37)
This SME was first derived in [Bel87]. The density operator ρ(t) gives at any time the knowledge of the observer about the state of the system consid-ering also the measurement record y(t); this means that this is a conditional state.
The first part of equation (2.37) can be recognized as a typical part of a Lindbladian and corresponds to eq. (2.4) when ˆL=√k ˆX. The second part of the evolution is the one that depends on the measurement record.
Equation (2.37) gives the evolution of the system at any time: in each intervaldt the measurement is performed and gives a certain result dy which is not deterministic because of the dependence on the stochastic quantity dW . Then the state of the system at the next step can be calculated with equation (2.37) using
dW =√8k(dy − h ˆXidt). (2.38)
This can be used as a numerical method to compute ρ(t), but has also a clear physical interpretation.
The evolution of the density matrix depends on a random quantity; given a certain sequence ofdW s we say that the ρ(t) derived from it is a quantum trajectory (cfr. [Car93]), and the sequence is called realization or unravelling of the noise. The probability of each trajectory is the probability to obtain the corresponding noise unravelling. Equation (2.37) describes the evolution of the conditional state (see section 1.1.5 on page 5), since it depends on the measurement outcome throughdW , as stated in equation (2.38). If one wants the evolution of the unconditional state, the average on all possible noise realization must be taken. We will show in the following section that hhρ dW ii = 0 (properties 14 on page 34), so in this case equation (2.37) is left only with the first part and become a standard Markovian master equation (equation (2.3)).
Wiener processes A Wiener process is defined as a continuous-time stochas-tic process W(t) that has W (0) = 0, the increment W (s) − W (t) is Gaussian with mean0 and variance s − t and increments for non-overlapping time in-tervals are independent (see Ref. [Gil96]). The probability density for W(t)
is P(W, t) = √1 2πtexp −W 2 2t . (2.39)
Intuitively W(t) is continuous, but non differentiable, as it is build by a sequence of steps; it is however possible to define the Wiener increment
∆W (t) ≡ W (t + ∆t) + W (t). (2.40)
Also ∆W is distributed according to a Gaussian with mean value 0 and variance∆t. It it clear that this ∆W has the same properties as the quantity defined in equation (2.32).
According to the Itô rule for stochastic calculus in the limit of ∆t → dt the Wiener increment ∆W → dW and (∆W )2 → (dW )2 = dt as assumed
in the previous section. Let us now prove these properties in detail. Properties 13. (dW )2 = dt
Proof. Performing a simple change of variable, the probability distribution for the stochastic quantity x= (∆W )2 is
P(x) = exp−x
2/2∆t √
2π∆tx . (2.41)
Using double angle bracket hh·ii to indicate the average on all possible noise realizations, the mean value and variance of(∆W )2 are
hh(∆W )2ii = ∆t Var[(∆W )2] = 2(∆t)2. (2.42) To see what happens in the continuum limit, let the total time interval t be divided equally in N intervals of length ∆t = t/N . For each interval we have a Wiener increment∆Wn≡ W ((n + 1)∆t) + W (n∆t).
The sum of the squared increments
N −1
X
n=0
(∆Wn)2 (2.43)
represents a random walk of N steps, with each step described by a proba-bility distribution with mean value t/N and variance 2t2/N2 as observed in
equation (2.42).
Applying the central limit theorem, in the limit of N → ∞ the probability distribution of the sum of squared increments becomes a Gaussian with the same mean value and variance, that is respectively t and2t2/N . But in the
same limit the variance →0 and the sum becomes t with certainty. This is quite remarkable: ∆W (t) is a stochastic quantity that varies randomly, but its square (∆W (t))2 is not random at all.
Since this is true for every time interval (0, t) we can identify the differ-entials(dW )2 = dt.
Properties 14. hhdt dW ii = 0 and hhρ dW ii = 0.
Proof. The first property is easily shown as the mean value of ∆W is 0. For the second property let us consider a simpler form of a stochastic differential equation
dy = a dt + b dW, (2.44)
The thesis becomes then hhydW ii = 0. From the evolution of y(t) we have
y(t + ∆t) = y(t) + a∆t + b∆W (t). (2.45)
This means that y(t) depends on ∆W (t−∆t), but not on ∆W (t). Since W (t) at different time are independent, it follows that the average of y(t)∆W (t) must be zero.
In conclusion we can sum up the basic rules of Itô calculus as
(dW )2= dt, (2.46a)
(dt)2 = dt dW = 0. (2.46b)
Multiple observers and inefficient detection Let us generalize the previous case when the same system is measured by two observers Alice and Bob. Suppose that Alice monitors observable ˆX with strength k1 and
Bob monitors ˆY with strength k2, suppose also that they do not
communi-cate between each other. Since Alice knows that Bob is monitoring ˆY , but she has no information about the outcomes of his measurement, it is clear that, from her point of view, Bob’s measurement induces only the dynamics −k2[ ˆY ,[ ˆY , ρ1]], where ρ1 is the state of the system according to Alice. This
means that the total evolution of the system according to Alice is (see for details Ref. [DDZ04]) dρ1 = −k1h ˆX,h ˆX, ρ1 ii dt − k2h ˆY ,h ˆY , ρ1 ii dt+ +p2k1 ˆXρ1+ ρ1X −ˆ 2 h ˆXi1ρ1 dW1, (2.47) where h ˆXi1 ≡ Trh ˆXρ1 i and dr1 = h ˆXi1dt + dW1 √ 8k1 (2.48) is the measurement record of Alice.
The total evolution for Bob is symmetrical, let us call ρ2 the state of the system according to him anddr2 his measurement record.
