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Phase Sensitivity in Quantum Interference Measurements
Oleg ˙Zero
Promoter: Prof. Gunnar Bj¨ork
Master dissertation submit...
Abstract
The sensitivity limit of two-mode interferometry is found in the presence of photon-loss, assuming strict
amonut ...
Foreword
This diploma thesis has been written within the Erasmus Mundus Master of Science in Photonics Programme
(EMMP) at...
Contents
1 Quantum Interference 4
1.1 The Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
Introduction
Interferometry is a technique, which possesses a truly supreme potential when high precison of measurements
i...
Chapter 1
Quantum Interference
When speaking about interferometers, we usually think about various optical setups involvin...
1.1.2 Phase Shifter
In Quantum Optics, the phase shift is fundamentally related to the time-evolution of the state (Ref. [...
1.2 The Interferometric Setup
The interferometric setup considered in this work is the Mach-Zehnder (MZ) interferometer. T...
1.2.2 Lossy Case
In realistic situations, the interferometer can never be absolutely lossless. In particular, photons can ...
After tracing the loss modes we obtain the reduced density matrix:
ˆρ
(1,2)
int = Tr3,4(ˆρint) = =
n1,n2
n1,n2
1, 2
γ( 1, ...
Chapter 2
Beyond Classical Meausurements
Now, after we have seen of how the Mach-Zehnder interferometer is treated by quan...
Here, ¯N is the average number of detected photons in the coherent mode. The inequality (2.2) defines the
sensitivity limit...
Remark on the Phase Super-Resolution
The phase super-sensitivity is often associated with another phenomenon: phase super-...
The Classical Fisher Information
The accurateness of an estimator is characterised by the mean square error:
mse(ˇθ) = (ˇθ...
X = {X1, X2, ..., Xi, ...}. If the operators commute, we can assign each estimator an operator ˇθ → ˆΘ,
such that ˆΘ|ˇθ = ...
The matrix elements of ˆL can be computed by writing ˆρ(θ) in its eigenbasis and using eq. (2.15), according
to:
[ˆL]ij =
...
schemes? Can we still beat the SQL using photon counting detectors and ‘normal beam splitters’? If yes,
then how?
2.3.2 Re...
When the number of measuremets is large M 1, then by virtue of the Central Limit Theorem, the
equation (2.22) becomes Gaus...
Chapter 3
Searching for the Optimum
We have finally gathered all tools needed to understand and correctly interpret all pos...
3.1.3 Possible Solutions
Primarly, we could ask why not replacing a high-N-photon state with more states of lower N and in...
3.2.1 2-photon State Modelling
Let us assume we have an arbitrary 2-photon input state:
|ψ in = x2,0|2, 0 + x1,1|1, 1 + x0...
can further be limited to Ri = [0, π/2]; Zi = [0, 2π); i = 1, 2, (3) in order to reduce the computation time.
Finally, the...
We collect the emerging photons at each branch independently, therefore
I(coh)
(ϕ0) = I(coh)
a (ϕ0) + I
(coh)
b (ϕ0). (3.1...
Figure 3.3: The sensitivity of 3-photon states against losses. Left: the symmetric case (η), Right: the asymmetric
case (µ...
possible explanation6
to this fact could be that if N is set odd, there exists no |N/2, N/2 component inside
the interfero...
3.5 If Accessing the Inaccessible...
Finally, the publication by Lee, et al. has encouraged7
me to investigate just one mo...
Consequently: I(X4) = 0, but ˜I(X4) = 0. Exactly the same situation happens for the (0,1) outcome.
Figure 3.5: Left: Fishe...
Discussion
Two major tasks have been realised in this work. First of all, two existing statistical methods were com-
pared...
Bibliography
[1] Gunnar Bj¨ork and Jonas S¨oderholm:
“The Dirac-notation in quantum optics”,
ICT Department, KTH Electrum ...
[11] T. Kim, J. Shin, Y. Ha, H. Kim, G. Park, T. G. Noh, C. K. Hong:
“The phase-sensitivity of a Mach-Zehnder interferomet...
[24] Leslie E. Balentine:
“Quantum Mechanics a Modern Development”
Simon Fraser University,
World Sicentific Pbs. (1998).
o...
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  1. 1. Phase Sensitivity in Quantum Interference Measurements Oleg ˙Zero Promoter: Prof. Gunnar Bj¨ork Master dissertation submitted in order to obtain the academic degree of Erasmus Mundus Master of Science in Photonics Academic year 2009-2010
  2. 2. Abstract The sensitivity limit of two-mode interferometry is found in the presence of photon-loss, assuming strict amonut of resource in each measurement. Since a strong emphasis is put on practical realisation possibility, a comprehensive discussion on the statistical methods and their corresponding potenial for implementation is made. Furthermore, by deveoping of a computer algorithm, the optimal qunatum states are found for N = 2, 3 number of photons, although the method can be extended to account for an arbitrary N. The performence these states is, then, compared with the standard quantum limit, in both cases of having loss in one or in two interferometer’s arms. Finally, the method is verified and the case of having access to a ‘surplus’ qunatum information concerning particular loss instances is given.
  3. 3. Foreword This diploma thesis has been written within the Erasmus Mundus Master of Science in Photonics Programme (EMMP) at Kungliga Teknisha H¨ogskolan (KTH) in Stockholm. First of all, I would like to thank Prof. Gunnar Bj¨ork, the promoter of this thesis, for his supervision and the time he has devoted in order to guide me. Furthermore, I would also like to thank to all professors of the EMMP commission, especially to Prof. Irina Veretennicoff, who introduced me into the World of Quantum Optics and to Prof. Roel Baets and Prof. Min Qiu who agreed on this thesis topic. Last, but not least, I would like to thank my dearest parents for their support, trust and belief in me, which without this thesis would have never been written. Oleg ˙Zero, Stockholm 2010 1
  4. 4. Contents 1 Quantum Interference 4 1.1 The Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 The Description of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Phase Shifter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.3 Beam Splitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 The Interferometric Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.1 Lossless Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2.2 Lossy Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 Beyond Classical Meausurements 9 2.1 What is the Phase Sensitivity? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 The Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 The Quantum Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.3 Between the SQL and the Heisenberg Limit . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Quantum Interference Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.1 Understanding the Output – The Estimation . . . . . . . . . . . . . . . . . . . . . . . 11 2.2.2 Quantum Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Criterion for Assessing the Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.1 The Canonical Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.2 Realistic Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Efficiency vs Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.1 Building the Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.2 The N00N State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4.3 Robust States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Searching for the Optimum 17 3.1 Global Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.1 The Problem of Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.2 The Problem with Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.3 Possible Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Preliminary Assumptions for Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2.1 2-photon State Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.2 3-photon State Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.3 The Implementation (Fock States) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.4 Coherent State Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.4 The Confirmation of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.5 If Accessing the Inaccessible... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2
  5. 5. Introduction Interferometry is a technique, which possesses a truly supreme potential when high precison of measurements is a required. After the invention of laser by Shawlow and Townes and its demonstration by Maiman in 1960, interferometry has found a great number of applications, where metrology, astronomy, surface scanning, remote sensing, telecommunications are just few examples (Ref. [13]). However, as the demand for precision, especially in scientific areas such as gravitational wave detection and quantum information, constantly increases, the questions of the the sensitivity limits become more and more important. It turns out that, even in the absence of any external sources of noise, classical light is, in its nature, a stochastic process, which imposes bounds to precision of phase measurements. Although, these bounds cannot be overcome by any classical means, quantum optics promises the sensitivity beyond that limit. This promise, also known as phase super-sensitivity, became a research area, whose exploration started around fifteen years ago. Various methods have been proposed by different groups (Refs. [5, 7, 11, 21, 22, 23]) in order to reach the quantum improvement. However, no super-sensitivity has been experimentally demonstrated with using quantum states of more than N = 4 photons until now (Ref. [6, 7]). As the phase sensitivity is closely related to specific quantum states of light used for the measurement, the ‘race’ to find the optimal states has begun. In 2007, P. Meystre and H. Uys (Ref. [2]) were the first who conducted a systematic search for such states, although, at that stage no losses were accounted in the mathematical model. In 2009, Demkowicz-Dobrza´nski, et al. (Ref. [3]) found the optimal states under the presence of loss. However, the specific assumptions they made concerning both the accessable quantum information and detection strategy, gave their research purely theoretical character. In this thesis work, the phase sensitivity limit is investigated with a strong empahsis put on practically implementable methods and strict resource limits (photons) used in the measurement. The optimal states are being found, by extending the methods developed by Ref. [2] to account for the presence of photon loss. Contrary to Ref. [3], we have assumed non-optimised detection scheme. Still, certain practical issues, such as the experimental cost of preparing the quantum states and non-ideal detection network, although mentioned, are not particularly focused on. The thesis is organised as follows: In Chapter I, a very brief introduction (or rather revision) of the some useful quantum optical concepts is made. The mathematical description of qunatisized light and relevant optical components is given and the interferometric setup is presented using the quantum ‘language’. Chapter II defines the phase sensitivity and introduces statistical methods that are indespensable to analyse the measurements’ outcomes. Two important statistical theories are presented and compared in the context of the detection strategy that helps us to evaluate the criterion for optimisation. Having both, the system and the statistical tools defined, the optimisation procedure and the results are presented in Chapter III. Finally, a concluding discussion is made. Last, but not least, it should be noted at this stage that with the strongest and honest aspiration to discover something new, I was not the first person, who pursued this idea. Lee, et al. (Ref. [4]) published independently, and unknown to me, almost identical the result in December 2009. Although I was not aware of their publictaion until almost the very end days of my research, still the confrontation of my results shows a perfect match between our data. See Section 3.4 for more details. 3
  6. 6. Chapter 1 Quantum Interference When speaking about interferometers, we usually think about various optical setups involving components, such as: mirrors and beam splitters. When speaking about light interference, it is natural, for at least most of us, to picture the light as a coherent and continues stream of many photons, likely being created with a laser and whose wave character allows it to produce the beautiful fringe patterns. In Quantum Interferometry, however, we are more concerned of sending single photons “one by one” or “group by group”. We arrange photons in particular superpositions of states and send them through an interferometer. Finally, what we detect are particular events or outcomes, rather than intensity variations or output power changes. In order to be able to describe the whole process, we need to refer to a more general theory, which is the Quantum Optics. In this chapter, we shall see a short, but complete description of the interference in terms of this theory. 1.1 The Mathematical Model 1.1.1 The Description of Light The state of light being in a certain mode can be represented as a superposition of the Fock states: |ψ ς = ∞ n=0 cn|n ς, cn ∈ C ∧ |n ς ∈ Hς, (1.1) where cn’s are the probablility amplitudes, n refers to the number of photons in the mode ς and Hς is the Hilbert space of that mode. As a paricular superposition of Fock states, it is also possible to express the coherent state, which is the quantum optical representation of the light, form an ideal laser: |α = ∞ n=0 αn √ n! |n , α ∈ C. (1.2) Finally, in order to be able to describe the common state at both interferometer’s branches, we need to account for the two modes representing each arm. We can do so using the tensor product to define the ‘joint’ Hilbert space and kets: |ψ = ∞ na=0 nb=0 cna,nb |na, nb , (1.3) where: |na, nb = |na ⊗ |nb ∈ Ha ⊗ Hb = Ha,b. 4
  7. 7. 1.1.2 Phase Shifter In Quantum Optics, the phase shift is fundamentally related to the time-evolution of the state (Ref. [1]). What is more, the light mode itself is analogous to quantum harmonic oscillator. Therefore, the Hamiltonian responsible for the phase shift is the harmonic oscillator Hamiltonian: ˆH = ω ˆa† ˆa + 1 2 = ω ˆn + 1 2 (1.4) The unitary operator responsible for the phase shift transformation can be found by solving the time- dependent Schr¨odinger equation using this Hamiltonian. In the Fock state basis, this operator reads: ˆUϕ = e−i ˆHt/ = e−iωtˆn = e−iϕˆn (1.5) Note that, the ω/2 is a common factor and hence it may be skipped without any consequences. To ilustrate, let us take the eq. (1.1) and use this operator: ˆUϕ|ψ = ∞ n=0 cne−iϕˆn |n = ∞ n=0 cne−iϕn |n (1.6) The Fock states are the eigenstates of the photon number operator ˆn, which is why we can write ˆn|n = n|n . As we can see, different Fock components existing within the superposition acquire different phase shifts that are directly proportional to the photon number. 1.1.3 Beam Splitter In most general terms, a beam splitter is a four-port1 that combines two optical modes and transforms them into another two modes. From the quantum point of view, this situation corresponds to having two coupled harmonic oscillators , where the coupling is described by the following Hamiltonian (Ref. [1]): ˆH = κ(ˆa†ˆb + ˆaˆb† ) (1.7) The operators: ˆa, ˆa† and ˆb, ˆb† refer to the first and second beam splitter’s inputs respectively and the κ is related to the coupling modes’ interaction time giving us the coupling strength Again, we would be interested in finding an explicit form of the unitary operator that is responsible for the beam splitter’s tranformation: ˆUBS = e−iκt(ˆa†ˆb + ˆaˆb† ) (1.8) However, since this operator would not operate in the its eigenbasis, and therefore, a way to handle this problem is to define the following subsititution of creation operators: ˆa† BS −−→ ˆa† cos κt + ˆb† sin κt ˆb† BS −−→ ˆb† cos κt − ˆa† sin κt (1.9) This set of “equations” can be derived from the time-evolution of the annihilation operators in the Heisenberg picture, using an equivalent Hamiltonian2 : ˆH = i κ(ˆa†ˆb − ˆaˆb† ). Substituting: √ ξ = cos κt and √ 1 − ξ = sin κt, when ξ ∈ [0, 1], we can also state (1.9) the more convinient form: ˆa† BS −−→ ˆa† √ ξ + ˆb† √ 1 − ξ ˆb† BS −−→ ˆb† √ ξ − ˆa† √ 1 − ξ (1.10) Now, we can clearly refer ξ to the beam splitter’s transmittivity. If we set ξ → 0.5, we obtain the transformation for the 50:50 beam splitter. Similarly, setting ξ → 0 turns our beam splitter into a perfect mirror. 1Although, in this context, the beam splitters are used as two-ports 2This Hamiltonian represents exactly the same physical truth. What is more, its form also preserves the hermicity. Therefore, the choice between (1.7) and this Hamiltonian is purely the matter of convinience. 5
  8. 8. 1.2 The Interferometric Setup The interferometric setup considered in this work is the Mach-Zehnder (MZ) interferometer. This setup is a fairly simple, but also a powerful tool for performing phase measurements. As we know, in its purest form, the MZ consists of two branches or arms: A and B, from which at one, the light is induced the relative phase shift: ϕ and the other one serves as the referrence beam. Both arms are coupled through beam splitters (BS) and the relative phase shift is induced by the phase shifter (PS). The PS may in fact refer to just any material or region of space that changes the optical path difference between the two arms. Figure 1.1: The schamtic of the Mach-Zehnder interferometer. Traditionally, for the MZ interfometer3 , we can discriminate three specific regions. Region I is the interferometer’s input stage, which is where the light enters the setup. The first beam splitter, then splits the mode into two, after which we define the region II – the MZ’s internal part. Finally, the second beam splitter recombines the modes, after which we define the interferometer’s output – the region III. 1.2.1 Lossless Case When no loss is present, then our MZ is just what we can see on the Fig. 1.1. We assume an arbitrary pure input state |ψin of the follwoing form: |ψin = ∞ n1=0 n2=0 xn1n2 |n1, n2 (1.11) When light propagates through the interferometer, the state (1.11) undergoes the following three transfor- mations (Ref. [2]), in the consecutive order: |ψout(ϕ) = ˆUBS2 ˆUϕ ˆUBS1 |ψin (1.12) Under these transformations, we obtain the output state |ψout(ϕ) , which is also a pure state but, whoose output components’ amplitudes interfere with each other in various ways |ψout(ϕ) = ∞ n1=0 n2=0 yn1n2 (ϕ)|n1, n2 (1.13) This interference gives rise to phase-dependent probablility distributions for the outcomes, which are the pos- sible numbers of photons, leaving the interferometer at each branch: (n1, n2). Analysing those distributions, we learn about the phase. Finally, it should also be mentioned that most often, the total number of particles N used in a single measurement is known and fixed. In these cases: (n1, n2) ≡ (k, N − k) and the outcomes can be identified by the single number k ∈ [0, N] ⊂ N. 3It is also true for other setup configurations, such as: the Michelson, Fabry-P´erot or Sagnac interferometer. Although the geometry of those setups are arragned differently, one can still discriminate between the three parts and the whole analysis can be carried out with the same methods. The choice of the MZ originates from the simplicity of its form. 6
  9. 9. 1.2.2 Lossy Case In realistic situations, the interferometer can never be absolutely lossless. In particular, photons can be lost due to the interferometer’s imperfections, and due to possible loss existing in the sample. Whenever any of these mechanisms is present, the possibility one or more photons, needs to be accounted in the mathematical model. Figure 1.2: The Mach-Zehnder interferometer with losses. The Model of Loss To account for the loss, our MZ is now equipped with two additional fictous beam splitters (Refs. [3, 8, 9, 10]): BSη 1 and BSη 2 with variable transmission parameters: η1 and η2, as can be seen on the Fig. 1.2. The fictous beam splitters mimic losses by sending photons to the loss modes (3) and (4), which can be understood as an extention to interferometer’s Hilbert space. Again, we use the tensor product to define the joint space: |n1, n2; 1, 2 ∈ H1,2 ⊗ H (loss) 3,4 (1.14) For any ket, the numbers: 1 and 2 denote how many photons are lost at the first or the second arm, respectively. The creation operators to the loss modes are: ˆa† η and ˆb† η, and the transformation can be defined in similar way to (1.10):    ˆa† BSη 1 −−−→ ˆa†√ η1 + ˆa† η √ 1 − η1 ˆb† BSη 2 −−−→ ˆb†√ η2 + ˆb† η √ 1 − η2 (1.15) In any real situation, we do not have any means to trace the lost photons. Therefore, we must literally trace-out the two additional modes, which leaves us with a mixed state that is described by the reduced density matrix. The “Families” of States Let us have a closer look at what happens to the state in the inerferometer. After the 1st beam splitter, the input state |ψin becomes the internal state, which undergoes the phase shift and the loss-transformation. It can be expressed as follows: |ψint(ϕ) = n1,n2 1, 2 γ( 1, 2) n1,n2 (ϕ) |n1, n2; 1, 2 , (1.16) where n1 and n2 are limited by the total number of photons N, and 1, 2 are limited by n1, n2. The complete density matrix of that state is: ˆρint = |ψint(ϕ) ψint(ϕ)| = = n1,n2 n1,n2 1, 2 1, 2 γ( 1, 2) n1,n2 γ ( 1, 2) n1,n2 ∗ |n1 − 1, n2 − 2; 1, 2 n1 − 1, n2 − 2; 1, 2| (1.17) 7
  10. 10. After tracing the loss modes we obtain the reduced density matrix: ˆρ (1,2) int = Tr3,4(ˆρint) = = n1,n2 n1,n2 1, 2 γ( 1, 2) n1,n2 γ ( 1, 2) n1,n2 ∗ |n1 − 1, n2 − 2 n1 − 1, n2 − 2| (1.18) This expression looks rather complicated. Luckily, from this mixed state, we are able to discriminate the several groups of pure states, corresponding to each of possible configurations of lost photons: ( 1, 2). ˆρ (1,2) int = ˆρ0,0 =0 + (ˆρ1,0 + ˆρ0,1) =1 + (ˆρ2,0 + ˆρ1,1 + ˆρ0,2) =2 + · · · (1.19) Where = 1 + 2 is the total number of photons lost, defined for each instance. It is necessary to mention that since these states are pure, they are in the sense independent, meaning they do not interfere with each other. Because of this, each one may be treated separately when calculating the output state after the 2nd beam splitter. ˆU† ϕ ˆρ 1, 2 ˆUϕ = ˆρ (out) 1, 2 equivalent to: −−−−−−−−→ ψ( 1, 2) ϕ out (1.20) Finally, from now on, we will be referring to the state, where = 0 as the primary state as it is the state of the highest importance. Consequently, the two states, for which = 1 will be called secondary, and so on. The last family of states, where for fixed N; = N, will be referred to as the dead states as they are all vacuum states and thus phase-insensitive. Remark: The Order of Operations At this stage, it is perhaps useful to clarify that the order of the phase and loss transformations that belong to the region II does not matter. Indeed, it can be shown that whatever operation we start with, becuase of the tracing, the resultant reduced density matrix remains identical. 8
  11. 11. Chapter 2 Beyond Classical Meausurements Now, after we have seen of how the Mach-Zehnder interferometer is treated by quantum optics, including also the model of loss, we are ready to move on to the next stage. In this chapter, we are going to “forget” for a moment about the detailed analysis of the MZ itself. Instead, we will concentrate more on what it means to make a good measurement in terms of precision. Finally, we will search for how we can benefit from the quantum nature of light to increase the performance of our measurement beyond the classical limits. 2.1 What is the Phase Sensitivity? Let us think of our MZ as a black box, in which there exists some unknown parameter ϕ – a phase, and whose value we would like to know. The smallest detectable phase shift is said to define the interferometer’s sensitivity (Ref. [11]). The phase sensitivity is limited by various noise processes that prevent us from knowing the phase exactly. Therefore, a common and realistic criterion for assessing of how precisely we can know the phase is phase uncertainty, being defined as the square root of the variance, (Refs. [5, 11]): δϕ = (ϕ − ˇϕ) 2 (2.1) By ϕ we denote the true, actual phase shift that exists in the interferometer and ˇϕ is our guess on its value. Our natural goal is to minimise this difference. 2.1.1 The Classical Limit In almost all cases of classical interferometry, the most common “type” of light is laser light. In the ideal case, meaning the absence of any additional noise sources1 , the laser light can be modelled by quantum optical coherent state: |α , as already mentioned in Chapter I. Unfortunately, it is an intrinsic property of the coherent state that the number of photons detected is not exactly predictable, but follows Poissonian districution. This stochastic nature of light gives rise to fluctuations known as shot noise,2 . which is an ultimate limitation to the classical interferometry (Ref. [14]). The shot noise imposes the lower bound on the phase uncertainty: δϕ ≥ 1 √ ¯N (2.2) 1These can be any external factors that could add noise to the system, such us: vibrations, pumping current noise, amplified sponteneous emission, etc. 2The origin of these fluctuations can be explained in various ways. They may be seen as originating from the fact that each photon incident on a beam splitter is scattered independently, giving rise to the binomial distribution in the two modes that become anticorrelated. They may also be explained with the vacuum field, which enters the beam splitter on the other arm and introduces the noise (Refs. [13, 14]). Finally, it can also be associated with the detection process, which turns the light into photo-current (Ref. [14]). 9
  12. 12. Here, ¯N is the average number of detected photons in the coherent mode. The inequality (2.2) defines the sensitivity limit for the classical light, which is also known as the Standard Quantum Limit (SQL). 2.1.2 The Quantum Limit As we just saw, classical interferometry is limited by the SQL. Fortunately, the quantum nature of light offers us the possibility of going beyond that limit, if an appropriate quantum state of light and a detection strategy is used (Ref. [14]). Unfortunately, quantum interferometry also knows its limitations. Since the photon number operator3 and the phase operator of the form of (1.5) do not commute, they will be linked by an uncertainty relation. According to Ref. [5], the square root of the variance of the phase and the photon number of any quantum state are related by the following unequality: δϕδn ≥ 1/2 (2.3) If N is the upper limit to the number of photons used in a single measurement, then the photon number uncertainty is bounded by N and consequently, the minimum phase uncertainty is bounded by: δϕ ≥ 1 N (2.4) The last relation defines the ultimate limit to the precision of interferometry in general. Since it follows directly from the uncertainty principle, it is also recognised as the Heisenberg Limit (HL). 2.1.3 Between the SQL and the Heisenberg Limit Both limits to interferometry are presented on the log-log plot (Fig. 2.1), which clearly shows us the potential offered by quantum optics against the number of resource. Now, it is an important question: how we can physically explore this region? In other words, which states of light and which detection strategy will allow us to approach the Heisenberg limit or at least make it possible to go beyond the standard quantum limit? As we will see, the answer for this question is not straightforward, especially if we consider lossy schemes. Figure 2.1: The sensitivity scaling. N represents the number of photons used for a single trial and δϕ is the sensitivity. The region between the SQL and the Heisenberg limit defines the phase super-sensitivity. 3Actually, since we work with the fixed total number of photons N, it is more adequate in this situation to refer to the photon number-difference operator: ˆnab = ˆa†ˆa − ˆb†ˆb, where a and b refer to the two modes, respectively. 10
  13. 13. Remark on the Phase Super-Resolution The phase super-sensitivity is often associated with another phenomenon: phase super-resolution. Since both phenomena are often coexisting, it is important to distinguish between them. The phase super-sensitivity is essentially a quantum phenomenon, which is related to the improvement in precision of measurements (Fig. 2.1), if appropriate quantum states are used. Sometimes, it is accompanied with the so-called super- resolution that is referred to as N-fold increment to the variation of interference fringes with ϕ (Ref. [2]). On the other hand, phase super-resolution is not entirely a quantum property, as it has been alse demonstrated for the coherent light (Ref. [12]). 2.2 Quantum Interference Measurement In a classical measurement, what we basically detect is the output power in one or both of the interferometer arms4 that brings us the information about the phase. In quantum measurements, however, we rather deal with specific random outcomes or events and therefore we need statistical tools to be able to analyse the output. In most general terms, the problem of current analysis can be recognised as the inference of an unknown parameter (phase), based on the statistical data. This is the primary task for so-called Estimation Theory. However, depending on the particular statistical representation of the problem, we can adapt either the classical or quantum Estimation Theory. In the next few sections, we shall briefly present the essence of both approaches in terms of their con- straints and applicability. Based on that, we are going to agree on a common criterion for assessment for the performance of the quantum states and optimisation. 2.2.1 Understanding the Output – The Estimation First, let us now have a look at how to make the best possible accurate guess on the phase in the framework of the classical statistics (Refs. [15, 16]). Basic Assumptions of the Classical Estimation Theory We assume that we deal with a set of mutually exclusive events Xi, belonging to the space of all events X. This implies: ∀ Xi,j ∈X p(Xi ∪ Xj) = p(Xi) + p(Xj) ⇐⇒ i = j (2.5) where p(·) is the probability density function (pdf) defined for every event and thus: p i∈X Xi = i∈X p (Xi) = 1 (2.6) After perfoming M independent measurements (also called trials), we collect the data D, to be the following set: D = {X (1) i , X (2) i , · · · , X (m) i , · · · , X (M) i }. The pdf defined for every event is conditioned on some unknown (scalar5 ) parameter p(Xi|θ); θ ∈ P ⊆ R and our aim is to infer its correct value. In order to make the inference, we define an estimator6 ˇθ to be a decision rule, such that ˇθ = ˇθ(D, θ). Naturally, we demand our estimator to be as accurate as possible. 4Of course, there exist several ways to do it. We can, for instance, detect the light emerging from one arm. We may also detect power at both arms and then look at the difference, which “cleans” the signal from the background (classical) noise. Nevertheless, the quantum (shot) noise will still remain the limitation in any case. 5In a most general case it is possible to have a vector of parameters. However, in this work, considering only one parameter suffices. 6Estimators are ususally denoted as ˆθ. However, to distinguish them from quantum mechanical operators, they will be denoted as ˇθ. 11
  14. 14. The Classical Fisher Information The accurateness of an estimator is characterised by the mean square error: mse(ˇθ) = (ˇθ − θ)2 = ˇθ2 − ˇθ 2 variance + ˇθ − θ 2 bias2 (2.7) Note that, the mse expresses the deviation of the estimated value from the true value and hence it will later ‘fit’ to our definition of sensitivity. If the estimator is unbiased, which means that the last term in (2.7) is zero and our estimate is right on average, then the precision of the estimate is only limited by the variance. In such case the Cram´er-Rao theorem applies: var(ˇθ) ≥ 1 I(θ) (2.8) I(ˇθ) is known as the classical Fisher information and it is the property of the mathematical model itself, given by the following formula: I(θ) = ∂ ln p(Xi|ϑ) ∂ϕ θ 2 X|θ (2.9) The symbol X|θ is used to remind us that the average should be taken over the space of events, not over the parameter’s space P. If the estimator efficient, which means if it is unbiased and has the ability to infer θ with the lowest possible variance, then (2.8) becomes an equality. This situaltion is highly desirable, since with our estimator we are not only able to extract the maximum possible information form the system, but we also gain a direct correspondence between the Fisher information and the variance. Consequently, we may use I(θ) to find the uncertainty. In principle, efficient estimators are seldom found, or may not even exist at all, for certian problems. Fortunately, for the purpose of interferometry, such estimators have been found and we shall see an example, later on in this thesis. For now, the most important thing is to remember for which kind of problems or models this appoach can be applied. 2.2.2 Quantum Estimation In quantum mechanics, besides the classical uncertanty, our knowledge about the system is also limited by the fact that quantum systems are described by an intrinsically statistical theory, which impose certain restrictions to our measurements. According to this theory, we are only allowed to ‘look at’ those observables that are represented by hermitian operators and if more than one observable is considered, their corresponding operators are expected to commute, if the measurements are to be described by independent stochastical distributions. Based upon these restrictions, the “old”, classical Estimation Theory has been deliberately extended to its quantum successor. In the contrary to its classical counterpart, this “upgraded” theory promises us more than how to make a good estimate, but also, what is very important, it helps us to find a specific form of an operator that will actually allow us to extract the maximal information out from the quantum system (Refs. [17, 18]). Let us, therefore, have a quick look at what this theory offers. The Basic Assumptions of the Quantum Estimation Theory In quantum theory, the most general representation of a state is a density matrix, which belongs to a Hilbert space ˆρ ∈ H. Here, the state is assumed to depend on some scalar7 , parameter ˆρ(θ); θ ∈ P ⊆ R. The measurements are represented by a set of operators ˆΠi, whoose possible eigenvalues define the outcomes 7Similarly, the case analysed here is the single-variate case. In principle however, we may have a whole vector of parameters θ = [θ1, θ2, ..., θN ]. 12
  15. 15. X = {X1, X2, ..., Xi, ...}. If the operators commute, we can assign each estimator an operator ˇθ → ˆΘ, such that ˆΘ|ˇθ = ˇθ|ˇθ . The state |ˇθ ∈ H(P) belongs now to a space of all possible guesses and ˇθ has an interpretation of an estimate. In other words, the measurement means that we operate ˆΘ on an eigenstate |ˇθ and obtain the estimate ˇθ as a result. The joint conditional probability is defined as p(ˇθ|θ) = ˇθ|ˆρ(θ)|ˇθ , which is then used to minimise an associated cost function, analogically to minimising the rms in the classical case. Just as the density matrix is a generalised representation of a state, the POVM8 is the most general form for the representing a measurement. In this case operators ˆΠi not need to commute and they are being defined on every infinitesimal, disjoint sub-region of the parameter’s space ˆΠ(ˇθ; dˇθ); ˇθdˇθ ∈ P (Ref. [18]). For an arbitrary interval of the parameter space ∆ ⊆ P, the POVM is found by: ˆΠ(∆) = ∆ ˆΠ(ˇθ ; dˇθ ), (2.10) with ˆΠ(P) = ˆ1. Note that this statement is the quantum mechanical analogy to the classical case (2.6). The joint conditional probability is obtained by the trace: p(ˇθ ∈ ∆|θ) = Tr ˆρ(θ)ˆΠ(∆) . (2.11) This formula has an interprestation of the probability that our estimate ˇθ is found in ∆, when the system is in ˆρ(θ). Only in the special case, when our measurement is a projective measurement, the operator (2.19) becomes: ˆΠ(∆) = ∆ |ˇθ ˇθ |dˇθ , (2.12) where P |ˇθ ˇθ |dˇθ = ˆ1. The Quantum Fisher Information Again, we would like to know how we could maximise the accuracy of our measurement. The definition of the “quantum accuracy” is fortunately not much different from the classical one. If the measurement is unbiased, which means that ˆΠ(ˇθ) ˆρ(θ) = θ then our estimate is correct on average and its accuracy is dictated by the variance only: var ˆΠ(ˇθ) = P ˇθ − ˆΠ(ˇθ) ˆρ(θ) 2 p(ˇθ|θ)dˇθ (2.13) where p(ˇθ|θ)dˇθ = Tr[ˆρ(θ)ˆΠ(ˇθ; dˇθ)]. In this case, the quantum version of the Cram´er-Rao inequality becomes: var ˆΠ(ˇθ) ≥ 1 IQ(ˆρ(θ)) (2.14) with IQ(ˆρ(θ)) to be recognised as the quantum Fisher Information. With help of the so-called symmetric logarithmic derivative ˆL, that was introduced by Helstrom (Ref. [18]) and defined by the following relation: ∂ˆρ(θ) ∂θ = 1 2 ˆLˆρ(θ) − ˆρ(θ)ˆL . (2.15) We arrive at the formula for the quantum Fisher information: IQ(ˆρ(θ)) = Tr ˆρ(θ)ˆL2 . (2.16) 8The POVM stands for the Positive-Valued Operator Measurement. 13
  16. 16. The matrix elements of ˆL can be computed by writing ˆρ(θ) in its eigenbasis and using eq. (2.15), according to: [ˆL]ij = 2 λi + λj ∂ˆρ(θ) ∂θ ij (2.17) The λi,(j) are the eigenvalues of ˆρ(θ) and whenever λi + λj = 0 we set [ˆL]ij = 0. For now, the purpose of introducting these definitions may appear somewhat unclear or at least a little bit out of the context. However, in just next few sections, we are going to see why it is so important to understand the major assumptions of each of these two frameworks. 2.3 Criterion for Assessing the Performance If we think of the Fisher information, classical or quantum, as of some sort intrinsic quantity of the system whose extraction will determine the precision our measurement, it becomes natural to ask of which tools or methods we need in order to be able to extract the maximum of it. An efficient estimator is the answer for the classical case. The analogy existing in the quantum case is an appropriate POVM, also known as the canonical measurement. 2.3.1 The Canonical Detection What has been first showed by O. E. Brandorff-Nielsen and R.D. Gill (Ref. [19]) for pure states and then extended by A. Luati (Ref. [20]) to mixed states is the following inequality: var(ˇθ) ≥ 1 I(θ) ≥ 1 IQ(θ) (2.18) This chain clearly suggest that we should aim for extracting the quantum information. However, as just mentioned before, this strategy requires the canonical measurement to saturate the quantum Cram´er-Rao inequality (2.14). The specific form for such POVM has for long time been unknown, until it was first derived by B.C. Sanders and G.J. Milburn (Ref. [21]). Adapting the Fock-state basis and setting the estimate to be our phase shift ˇθ = ˇϕ, the form of this POVM (Ref. [5]) becomes: ˆΠcan( ˇϕ) = 1 2π | ˇϕ ˇϕ| , where: | ˇϕ = N n=0 ein ˇϕ |n, N − n (2.19) Unfortunately, the operator ˆΠcan( ˇϕ) must depend of what is to be measured and as also pointed out in the Ref. [5]: “Except in special cases, it is not possible to perform canonical measurements with standard optical equipment (photon counters and linear optical elements such as beam splitters)”. Furthermore, the meaurement of this fom ‘works’ only, if the interferometer is lossless, which means our output state is pure. As soon as even a tiniest loss is present in the system, our output results in a mixture of many pure states as indicated in Chapter I. According to Ref. [3], in order to extract the maximum possible information, we need one more piece of information about the system: we need to know exactly how many of, and where, the photons were lost before applying the appropriate POVM. Such information will project our density matrix onto one of the intrinsic pure states, which corresponds to a particular configuration of loss ( a, b); ˆρout Pr −→ ˆρ ( a, b) out . Only then we can apply the optimal POVM (2.19). Except for the difficulty to realise the POVM, prior knowledge about the loss causes an even greater technological issue. In each case, the requirement to know ( 1, 2) implies that we must measure the number of photons at each branch without absobing them! Such measurement can be qualified as a quantum non- demolition (QND) experiment, which although theoretically allowed, remains both practically undoable and likely unfeasable for the forseeable future. Since we are essentialy limited to using the “non-sophisticated” elements, such: beam splitters and photon counting detectors, the important question arises: What are the limitations to our precision if using realistic 14
  17. 17. schemes? Can we still beat the SQL using photon counting detectors and ‘normal beam splitters’? If yes, then how? 2.3.2 Realistic Detection An example of a realistic setup such as the lossy Mach-Zehnder interferometer has just been introduced in chapter I. Given the photon counters, we deal with the projective measurement in the Fock-state basis. Such measurement can be represented by the operator: ˆP = ˆa† ˆa;ˆb†ˆb : ˆP|na, nb = (na, nb)|na, nb (2.20) In this case, knowng the total number of photons N, we can still tell how many photons we have lost in total, but we are given no information where the photons were lost. What is more, this measurement strategy does not allow us to perform any adjustments during an act of a single measurement. Still we assume our detector’s network has an idel recognition between the events: |2, 0 will not be confused with |1, 0 , for instance, and the detector’s quantum efficiency is assumed to be unity – all loss is entirely in the interferometer. 2.4 Efficiency vs Robustness The direct consequence of the assumptions given in the previous section is that now, we arrive at classical estimation. Indeed, according to Helstrom (Ref. [17, 18]) having non-optimal, but well-defined operator and detection strategy, our estimation can be performed with the classical means. 2.4.1 Building the Estimator First of all, let us define the space of events X. Upon the measurement (2.20), our outcomes are the eigenvalues of the operator ˆP, which span the complete set XNph: XNph = 0 n=N n k=0 (n − k, k) ≡    (N+2)(N+1)/2 i=1 Xi    (2.21) The subscript Nph is used to remind us about the exact number of photons in one trial and the ‘reversed’ order of the first set-summation is a convention we use to indicate the ordering of the elements: (N, 0) ≡ X1, (N − 1, 1) ≡ X2 and so on. The occurence of each one of Xi’s is determined by the conditional probability density function p(Xi|ϕ), which also assumes an implicite dependence on loss and beam splitters’ strengths. The index i is used to distinguish between every outcome in X and the pdf’s can be calculated from the output density matrix ˆρout(ϕ). Based on the so-called Bayesian approach, we look for the posterior probabiliy p( ˇϕ|D) = p( ˇϕ| M m=1 X (m) i ), which is the probability of having the phase shift ϕ conditioned on all outcomes collected throughout mea- surement process. According to Bayes’ theorem: p( ˇϕ|D) = p( ˇϕ)p(D| ˇϕ) p(D) (2.22) p( ˇϕ) is the so-called prior probability distribution, which corresponds to our knowledge about the phase before any measurement is done. If nothing is known about the phase a priori, then p( ˇϕ) = 1/2π. The denominator is the a priori knowledge about the data and it is to act as the normalisation constant p(D) = P p( ˇϕ )p( ˇϕ |D)d ˇϕ . Finally, our parameter subspace is usually P ≡ [−π, π]. 15
  18. 18. When the number of measuremets is large M 1, then by virtue of the Central Limit Theorem, the equation (2.22) becomes Gaussian and according to Ref. [2], it can be approximated by: p( ˇϕ|D) ≈ p( ˇϕ|ϕ) = 1 N i∈X p(i| ˇϕ)Mp(i|ϕ) (2.23) Where N = P i∈X p(i| ˇϕ )Mp(i|ϕ) d ˇϕ is the normalisation constant. Equation (2.23) is the Maximum Likelihood Estimatior (MLE). Indeed, this function tells us how likely it is that our guess ˇϕ matches the true phase shift ϕ. By analysing the derivatives, it is proven in Ref. [2] that this function attains maximum when ˇϕ = ϕ and its width is equal to the variance. This means that in the asymphotic limit we have obtained the efficient estimator and therefore, we may use the (classical) Fisher information to optimise our system. Remark on Phase Ambiguities Last, but not least, it may happen that the likelihood function (2.23) possesses more than one maximum. If such situation happens, we face an ambiguity in recognising the phase, which is similar to super-resolution. This is, fortunately, a minor problem, since according to Ref. [5], a possible implementation of a feedback loop9 to the system or changing the input states from trial to trial will naturally solve the ambiguity, by concentrating the likelihood function around the true value. 2.4.2 The N00N State Before moving on to the next chapter to see how we can optimise the system, let us have a closer look at one, very specific example: the so-called N00N state: |N00N = 1 √ 2 |N, 0 + eiχ |0, N ; χ ∈ R (2.24) The name of this state is self-explanatory. If we look at its form, we quickly realise, why this state has been typed a good candidate for quantum interferometry. Indeed, if this state is the interferometer’s internal state, it will ‘pick-up’ the maximum possible phase shift (see eq. (1.5)), in principle, allowing us to perform measurements exactly at the Heisenberg limit. However, as soon as there exists loss in our system, loosing only one photon will cause a total decoherence of the system, degrading the state to a mixtrue that will exhibit no interference at all. To show this, let us define a one-photon loss operator ˆPη = [|N − 1 N| ⊗ ˆ1] ⊗ [|1 0| ⊗ ˆ1]loss and see what happens to the N00N state: ˆPη|N00N = 1 √ 2 |N − 1, 0 ⊗ |1, 0 loss (2.25) Under trace, we will discard the phase-information together with the loss-modes |1, 0 loss and our new state |N − 1, 0 will have no “partner” to interfere with. 2.4.3 Robust States We have just seen how fragile the N00N states are, even though their sensitivity to phase, is the highest possible. In practice losses will always occur and depending on each partiular case, there will be other states, whose sensitivity, although not as high, will be compensated by robustness. In the next chapter will shall see in more detail, which states are best for each case and how they are found. 9Such (electronic) feedback would possible re-adjust the auxiliary phase shift Φ placed at the second arm, after each trial. 16
  19. 19. Chapter 3 Searching for the Optimum We have finally gathered all tools needed to understand and correctly interpret all possible experimental outcomes. Now, we can use this knowledge to optimise the perfomance of our interferometer. As discussed in Section 2.4.1, the Bayesian estimator provides us the method to recognise the phase shift, whereas its ap- proximation, the MLE, becomes the efficient estimator in the asymtotic limit. Consequently, by maximising the Fisher information we immediately learn about the optimal performance. 3.1 Global Strategy As a first step, we should consider the global strategy, which means how we should proceed with our mea- surement in general. As presented in Fig. (2.1), the best precision allowed by physics is the Heisenberg limit. However, it is only attainable in the absence of loss, if the N00N or similar states are used (Ref. [2]). Essentially, the higher the number of photons N is used in a single trial, the more “beneficial” the measure- ment becomes. On the other hand, as N is being increased, it not only becomes much more cumbersome to prepare such state, and later, to appropriately detect its outcomes, but also the amount of loss we can tolerate quickly decreases. 3.1.1 The Problem of Loss This fact can easily be demonstrated if we equate the sensitivity of the N00N state (Ref. [3]) with the one of the coherent state: δϕSQL = δϕN00N ⇐⇒ 1 √ Nη = 1 N ηN =⇒ η(N) = N 1 1−N . (3.1) Setting ¯N = Nη for the coherent state and transforming the equation (3.1), we obtain the function η(N) that tells us the minumum tolerable transmission, below which the N00N state performs worse than the classical light. For example, if using a 100-photon N00N state, we must ensure that: η > 95% and if N = 103 , then η > 99%. This shows that the improvement promissed by the quantum light is greatly inhibited, even if disarding any other source of error. 3.1.2 The Problem with Implementation In reality, besides the loss, one must also consider the technological drawbacks. Although these are greatly excluded from current discussion, one should be aware of the two important issues which originate from the technological side: The first one is the experimental cost of preparing the input state. And the second one is to guarantee the low error-rate detection of the outcomes. Both of these issues put further bounds to the maximum number of photons which we can afford to use in any one trial. 17
  20. 20. 3.1.3 Possible Solutions Primarly, we could ask why not replacing a high-N-photon state with more states of lower N and increase the number of repetitions m? Indeed, this is the exactly how the measurement is conducted in practice. However, as each trial is purely independent, the Fisher information becomes additive, which leaves us with the classical scaling against the number of trials: δϕ ∝ m−1/2 . Until now, no phase super-sensitivity has been experimentally demonstated using states of more than N = 4 photons (Ref. [6, 7]). Still, the experimental realisation in Ref. [6] was conducted in the so-called post-selecton paradigm1 , which may be regarded as an approximation to canonical measurements, since the inference is based only on the “successful events”. However, when certain “undesired” outcomes are dismissed a posteriori, the exact number of resources needed for certain precision becomes undefined, thus not showing us the true sensitivity limit. Another method proposed, was to make a single photon pass though the phase shifter several times (Ref. [7]), which results in a phase-shift proportional to the number of bounces. Nevertheless, the analogical solutions are also known in the classical interferometry2 , where the presence of a high-finesse cavity enhances the precision. It will, therefore, be more appropriate to refer the quantum multipass interferometry to its classical counterpart and let the super-sensitivity itself be discussed agreeing on a most fundamental setup. 3.2 Preliminary Assumptions for Optimisation Taking all these facts into account, the following work was focused on optimising the low-N-photon states in the context of loss (N = 2, 3). Concerning the loss, two important situations have been treated explicitely: • the symmetric loss: η = [η1, η2] ≡ η, which is the model for the loss existing equally in both arms • and the asymmetric loss: η = [η, 1], which could be the model for a lossy phase shifter or sample In every case, the coupling strenth of both beam splitters has been set symmetric: ξ = [ξ1, ξ2] ≡ 1/2, unless stated otherwise. The optimisation algorithm was implemented using the Wolfram Mathematica 7.0 software. For this purpose, the classical (and discrete) Fisher information (2.9) I(ϕ0) = i∈X 1 p(Xi|ϕ) ∂p(Xi|ϕ) ∂ϕ ϕ0 2 (3.2) has been rearranged for the purpose of computation: I(ϕ0) = i∈X p(Xi|ϕ0) p(Xi|ϕ0 + ∆ϕ) p(Xi|ϕ0)∆ϕ − 1 ∆ϕ 2 . (3.3) The ϕ0 is the phase at which we bias our interferometer3 . For sufficiently small step (∆ϕ = 10−6 ), the simulation results show that the Fisher information is independent from the value of ϕ0. Except for ϕ0 = 0, it can be set to an arbitrary value. 1The post-selection paradigm is also known as the adaptive measurements. We could agree, for instance, on rejecting every event, in which we know we have lost photons. This procedure, will obviously increase the overall precision, but the exact number of photons remains undefined. 2For instance: the Fabry-P´erot or Fizeau interferometers. 3It is in any way related to the estimator bias, which is equal to zero for efficient estimators. 18
  21. 21. 3.2.1 2-photon State Modelling Let us assume we have an arbitrary 2-photon input state: |ψ in = x2,0|2, 0 + x1,1|1, 1 + x0,2|0, 2 , x0,2, x1,1, x0,2 ∈ C. (3.4) After all transformations discussed in Chapter I, we arrive at the output density matrix ˆρout(ϕ), from we can discriminate six pure states, whose components’ complex amplitudes are: y ( 1, 2) na,nb (ϕ), with na, nb referring to the number of photons and ( 1, 2) specifying the particular loss instance. The probability associated with each component can be calculated taking p ( 1, 2) na,nb (ϕ) = |y ( 1, 2) na,nb (ϕ)|2 . Accounting for the measurement operator (2.20), our space of all possible measurement outcomes reads: X2ph = {(2, 0), (1, 1), (0, 2); (1, 0), (0, 1); (0, 0)} ≡ 6 i=1 Xi . (3.5) It is essential to note that certain outcomes, i.e. the outcome: (1, 0) can be obtained loosing one photon in either of the arms. Since we have no means to monitor the lost photon, these two situations become indistinguishable. On the other hand, we also know they are mutually exlusive, hence (2.5) applies and we must add the probabilities p(X4|ϕ) ≡ p((1, 0)|ϕ) = p (1,0) 1,0 (ϕ) + p (0,1) 1,0 (ϕ), (3.6) before puting them to (3.2). Similar situation happens for all other indistinguishable, mutually exclusive events, not only in the case of 2-photon states. 3.2.2 3-photon State Modelling For the 3-photon case, we assume to have an arbitrary 3-photon input state: |ψ in = 3 k=0 x3−k,k|3 − k, k , ∀k x3−k,k ∈ C. (3.7) Again, all transformations discussed in Chapter I apply, but this time, our outcome space X3ph contains more events: X3ph = 0 n=3 n k=0 (n − k, k) ≡ 10 i=1 Xi . (3.8) All other procedures remain the same, as before. 3.2.3 The Implementation (Fock States) Now, the task is to search for such input amplitudes x2ph = [x2,0, x1,1, x0,2] (or x3ph = [x3,0, x2,1, x1,2, x0,3]), so that I is maximised. For this purpose, we parametrise the input state with i, ∈ Ri, ζi ∈ Zi; i = 1, 2, (3) as follows: |ψI 2ph = cos 1|2, 0 + eiζ1 sin 1 cos 2|1, 1 + eiζ2 sin 1 sin 2|0, 2 , (3.9) |ψI 3ph = cos 1|3, 0 + eiζ1 sin 1 cos 2|2, 1 + eiζ2 sin 1 sin 2 cos 3|1, 2 + eiζ3 sin 1 sin 2 sin 3|0, 3 . (3.10) The parametrisation in (3.9) and (3.10) ensures the normalisation. The parameter space, defined to be BN=2,(3) def =   i=1,2,(3) Ri   ⊕   i=1,2,(3) Zi   ⊆ R4,(6) , (3.11) 19
  22. 22. can further be limited to Ri = [0, π/2]; Zi = [0, 2π); i = 1, 2, (3) in order to reduce the computation time. Finally, the simulation is run and for each value of η, the state giving the highest I is found and I is plotted against η (the symmetric case) or µ (the asymmetric case, see later). Remark on the Parameter Space Sampling In order to reduce the computation time even further, the space B is sampled, usually taking 5 points from each Ri and 4 points from each Zi’s. Consequently, we find an approximation to the maximal Fisher information, which is, nevertheless, a good approximation, since increasing the number of samples4 in B leads to a difference no higher than 12% with the respect to the previously obtained value of I. For clarity, we shall refer to the number of samples as s ≡ (s 1 .s 2 .(s 3 ).sζ1 .sζ2 .(sζ3 )), when later discussing the results. Remark on the Asymmetry Treatment When the losses are symmetric, by definition η may be interpreted as the overall interferometer’s transmis- sion. This is not exactly true, when the loss becomes asymmetric. According to Ref. [8], the influence of the asymmetry can be treated by adjusting the coupling strength of the first beam splitter ξ1: ξ1 = 1 η + 1 . (3.12) If ξ1 is adjusted as indicated above, the visibility of the classical signal can be restored. However, in this case, the overall transmittivity can be expressed in more intuitive way. Let us introduce a new variable µ, which we will call an efficiency: Ψ = (ξ1η)Ψ + (1 − ξ1)Ψ (3.12) = 2η 1 + η Ψ def. = µΨ. (3.13) Where Ψ be an input power injected to the first arm and Ψ = µΨ is the output. Now, we have a coefficient that is expresses the loss in manner that is independent from the asymmetry – “output/input”. 3.2.4 Coherent State Modelling When discussing the improvement, the SQL provides us a natural reference point. Let us, therefore, briefly ilustrate how the coherent state is accounted in our simulation. Since any coherent state modes are perfectly uncorrelated, given |ψcoh in = |α, 0 injected as the input, the output state is easily found by repeting all previous transformations: ψcoh out (ϕ) = αe−iϕ η1ξ1ξ2 − α η2(1 − ξ1)(1 − ξ2) ⊗ αe−iϕ η2ξ1(1 − ξ2) + α η2(1 − ξ1)ξ2 = βa, βb . (3.14) Accounting for (1.2), the probability of detecting k-photons in output modes a, b is: p(k|βa,b) = e−|βa,b|2 |βa,b|2k k! . (3.15) Since |α|2 has the interpretation of the photon number average ¯N, and since in our case only ¯N = 2, 3 photon states are considered, (3.2) can be well-approximated, by taking only first ten terms of (3.15) I (coh) a,b (ϕ0) = 10 k=0 1 p(k|βa,b) ∂p(k|βa,b) ∂ϕ ϕ0 2 . (3.16) 4Increasing the number of samples from (5.5.4.4) → (5.5.8.8) in the 2-photon symmetric case shows no resolvable difference between the results. In the asymmetric case, the same increment leads to 12% for µ = 0.62, which is the highest difference occur- ing between the two solutions. Further increasing of the sampling density shows that the solution is converging. Unfortunately, due to the limited computational power of the computer, the convergence of the 3-photon as not investigated. 20
  23. 23. We collect the emerging photons at each branch independently, therefore I(coh) (ϕ0) = I(coh) a (ϕ0) + I (coh) b (ϕ0). (3.17) Formula (3.17) provides us exactly the referrence we need. Finally, it is worth noting that I(coh) is also independent from ϕ, just as in the case of Fock states. 3.3 Simulation Results Knowing all the procedures exactly, we are now prepared for presenting the results. The following figures summerise the results for collected in each simulation. Figure 3.1: The sensitivity of 2-photon states represented by the Fisher information against losses. Left: the symmetric case I(η), Right: the asymmetric case I(µ). Each black line corresponds to a particular state, which is the optimum within a certain interval. The gray region: classical measurement. The red line: the N00N state. The dark-gray line: the x0 state. Fianlly, the dotted-green (contour) line represents the overall optimum, which defines the sensitivity limit in each case. Figure 3.2: The sensitivity of 2-photon states against losses. Left: the symmetric case (η), Right: the asymmetric case (µ). The dark-blue dots: the optimal states with (5.5.8.8); the bright-blue dots: the optimal state with (5.5.4.4); the red line: the N00N state, the gray line: the coherent state with ξ1 = 1/2; the black line: the coherent state with optimally adjusted beam splitter: ξ1 = 1/(1 + η). 21
  24. 24. Figure 3.3: The sensitivity of 3-photon states against losses. Left: the symmetric case (η), Right: the asymmetric case (µ). The dark-blue dots: the optimal states with (5.5.5.4.4.4) assuming the input amplitudes complex xin ∈ C; the bright-blue dots: the optimal state with (5.5.5.1.1.1), but fixing all amplitudes real xin ∈ R; the red line: the N00N state, the gray line: the coherent state with ξ1 = 1/2; the black line: the coherent state with optimally adjusted beam splitter: ξ1 = 1/(1 + η). 3.3.1 Discussion The 2-photon Case The optimisation algorithm shows the existance of several input states, whose sensitivity dominates the sensitivity of all the other input states within specific regions of loss (Fig. 3.1). The selected optimal states are essentially different in each case (symmetric/asymmetric loss), exept for in the two extreme cases: x0 = [1, 0, 0] and xN00N = [1, 0, 1] / √ 2. The latter input state corresponds to the internal N00N state (also achievable with [0, 1, 0]-input), which has been introduced in Section 2.4.2 as yielding the highest sensitivity. From Fig. 3.2, we can observe that the N00N state is selected ‘best’ in the low-loss regime: ηN00N,2 ≥ 0.65 (symmetric) and µN00N,2 ≥ 0.96 (asymmetric), attaining the highest Fisher information I = 4 if the interferometer is ideal. Shortly after reaching the threshold (ηN00N,2 or µN00N,2), its performance gets quickly degraded, which stands in agreement with Refs. [3, 9, 10], (see also eq. (3.1)). The existance of such high threshold given for the N00N state in the asymmetric case can be explained with the fact, that the N00N state is a symmetric state itself, hence might naturally be less tolerant to losses that are present only in one arm. Below ηN00N,2 (or µN00N,2), more robust states of lower sensitivity are found. And for extremely high losses η < 0.13 (and µ < 0.11), the x0 state becomes the optimum. Its sensitivity, in the symmetric case, coincides perfectly with the SQL-line, signifying the classical performance. In the asymmetric case, the state performs slightly better. The 3-photon Case First of all, the results obtained in this case indicate ‘a proper’ scaling of the Fisher information5 against the number of photons N. In either case (symmetric and asymmetric), when no loss is present, the Fisher information for the N00N state grows quadratically, whereas it remains linear for the coherent state. What is more, when losses are present Icoh is also linear vs. η or µ, provided that ξ1 is optimally adjusted. Most of the conclusions formulated for the 2-photon case ar also valid when N = 3, except for the fact, that a slightly different state (other than x0 = [1, 0, 0, 0]) is preferred in the high-loss regime, when the loss exists in one arm only. Similarly to when N = 2, the N00N state performs best when losses are low, although its regime is now bounded by new thresholds: ηN00N,3 ≥ 0.74 or µN00N,3 ≥ 0.83. Surprisingly, the region of the N00N state extands further, comparing to the previous situation, if the loss is asymmetric. The 5...and hence the uncertanty, since δϕ = 1/ I(ϕ) when saturating the Cram´er-Rao bound. 22
  25. 25. possible explanation6 to this fact could be that if N is set odd, there exists no |N/2, N/2 component inside the interferometer and therefore, it might somehow be easier for the odd-numbered N00N states to resist asymmetries. Finally, we should emphasise the importance of assuming the input amplitudes complex, before proceeding with the optimisation. As we can see on Fig. 3.3 (right), when the amplitudes are constrained real, the space B is being greatly reduced (dim B = 4, (6) → 2, (3)) and the “optimal” states then found may even exhibit sensitivity lower than that of the coherent state. 3.4 The Confirmation of the Method As stated in the Introduction, all optimisation methods developed to investigate the influence of photon- loss, which I was aware of throughout the whole time of my thesis work, were entirely based on the quantum Fisher information (Refs. [3, 9, 10]), thus, as discussed in Chapter II, not revealing us the realistic limit to the sensitivity of interferometers. The motivation for my work, therefore, was to find the sensitivity limit assuming a more implementable setup, which I have found by extanding the idea of H. Uys and P. Meystre (Ref. [2]) and choosing the classical Fisher information to build my algorithms upon. In the paper published in December 2009 by Lee, et al. (Ref. [4]), the authors proceed with the same reasoning and even extanding their algorithms to account for an arbitrary number of photons. Unfortunately, I have learnt about this publication too late in order to start my work anew and try to explore the unexplored, as I was already at the stage of summerising my results. The same publication has, on the other hand, granted me a great privilage and opportunity to confront my results, at least partially, with the work done by the scientists. Figure 3.4 presents a “common case”, which is used for the comparison: Figure 3.4: The sensitivity vs. the number of photons N used in one trial, for the loss-asymmetric case. This figure is the part of research conductd by Lee et al. (Ref. [4]). The markers with the numbers correspond to the data obtained by me and they are used for the comparison. The black numbers: the 2-photon case, the green-bold numbers: the 3-photon case. As we can observe from the Fig. 3.4, the position of the points calculated by me, match exactly the positions calculatd by Lee, et al. for the same points. Consequently, I take this agreement anyhow gives me a feeling of confidence that I was, indeed, on the right track. 6This is purely a hypothesis which requires verifictaion. 23
  26. 26. 3.5 If Accessing the Inaccessible... Finally, the publication by Lee, et al. has encouraged7 me to investigate just one more case. In Chapter II, when discussing the optimal measurement strategy, we mentioned that whenever the interferometer is lossy, the output state results in a mixture and the only ‘possibility’ for us to be able to perform the optimal measurements, is to somehow monitor the losses. It was also emphasised that such monitoring is practically impossible to construct. However, it is still interesting from the theoretical point of view, to see of what impact this information might have on the interferometer’s precision. Loss Monitoring and Distinguishability of Events Let I(Xi) be one of the terms of the sum I(X) = i I(Xi) (eq. (3.2)), corresponding to a particular experimental outcome Xi that is a result of the occurence of ˜Xi,1 and ˜Xi,2. It was indicated in section 3.2.1 that if ˜Xi,1 and ˜Xi,2 are mutually exclusive, indistinguishable events (2.5), we should add their probability functions and calculate the Fisher information (3.2) that is based on their sum: I(Xi) = I ˜Xi,1 ∪ ˜Xi,2 (3.18) Assuming now that we are able to monitor the loss, the events ˜Xi,1 and ˜Xi,2 are still mutually exclusive, but now, they become distinguishable. Since the Fisher information is calculated for the visible outcomes, we have: ˜I(Xi) = I ˜Xi,1 + I ˜Xi,2 (3.19) In general, we expect: ˜I(Xi) = I(Xi), which implies that depending on the distinguishability of events, we will arrive at different Fisher information in each case. The question now arises: How large the difference would be: ∆I = |I(X) − ˜I(X)|? Due to the large number of degrees of freedom, even for N = 2 the answer seems to be rather difficult to find analytically. However, inspecting the formula (3.2), we see that the Fisher information depends on the derivatives of each pdf with the respect to the phase p (Xi|ϕ). What is more, each derivative is squared, which makes I independent from sing of p (Xi|ϕ). Therefore, we expect that if a certain state is found, such that under losses, it ‘creates’ two events of the opposite probability slopes, then depending on wheather we calculate the Fisher information according to (3.18) or (3.19), we should arrive at ∆I = 0. The epsilon-State Let us consider a 2-photon state. Particularly, let us concentrate on two loss-origined events: |1, 0 (⊗|1, 0 loss) ≡ ˜X4,1 (3.20) |1, 0 (⊗|0, 1 loss) ≡ ˜X4,2 (3.21) The event (1, 0) ≡ X4 = ˜X4,1 ∪ ˜X4,2 is the observable outcome corresponding to |1, 0 output state. Now, let us consider an example to be the ε-state, defined as the following input state: |ψI(ε) = 1 √ 2 eiε |2, 0 + |0, 2 (3.22) Setting ε = π/2 we obtain a state, whose output pdf’s p (1,0) 1,0 (ϕ) and p (0,1) 1,0 (ϕ) are equal in amplitude, but being exactly out-of-phase. Hence: p(X4|ϕ) = p (1,0) 1,0 (ϕ) + p (0,1) 1,0 (ϕ) = const. (3.23) 7The reason for undergoing this investigation was due to my initial misinterpretation of their paper. I have, basically, misunderstood the notation used by the authors and hence postulated that they have not accounted for the fact that certain experimental outcomes apear indistinguishable to us. I was wrong. Nevertheless, I have decided to include these results as, in my opinion, it still might be interesting to observe of what happens if we are able to monitor the losses (as implicitely assumed in Ref. [8]). 24
  27. 27. Consequently: I(X4) = 0, but ˜I(X4) = 0. Exactly the same situation happens for the (0,1) outcome. Figure 3.5: Left: Fisher information obtained for |ψI (ε) vs. ε with ξ1,2, η1,2 = 1/2. I(X, ε) is calulated assuming the indistinguishability of outcomes, whereas ˜I(X, ε) represents the information when the loss is monitored. The case, in which ε = 0, 2π corresponds to the internal N00N state, which gives no sensitivity under losses, thus I(X, 0) = ˜I(X, 0). ε = π corresponds to the interal eiπ |1, 1 state, who experiences no relative phase shift, but a global phase factor, hence it does not yield any sensitivity at all. Right: The probability density functions of the ˜X4,1 and ˜X4,2 outcomes that originate from loss. As we can see, the probability variations are exactly out-of-phase and the highest difference between them occurs at ε = π/2 and 3π/2. Conclusion As we can see in Fig. 3.5, for ε = π 2 , (or 3π 2 ), for this case, we obtain the difference as high as ∆I(X, π 2 ) = 2I(X, π 2 ). This demonstrates the fact, that if we had an ability to monitor the losses, our sensitivity becomes improved. This improvement is expected to be highest in the region of moderate-loss, as the origin of this difference follows directly from the presence of losses. Furthermore, it is also expected that for higher N, this difference would become more significant, as the number of possible loss-desent events will grow. Finally, if the optimisation procedure is based under the distinguishability assumption, it is almost certain that different states would be evaluated optimal within certain intervals. 25
  28. 28. Discussion Two major tasks have been realised in this work. First of all, two existing statistical methods were com- pared and evaluated, focusing strongly on their applicablility when it comes to practically implementable measurements and accessible quantum information. Next, agreeing on a common criterion, the sensitivity limits were investigated assuming photon losses, existing equally in one or both interferometer’s arms. By developing a computer algorithm, the optimal quantum states were found in each case, assuming both the existance of realisable measurement schemes and no adaptive measurements for N = 2, 3 photon states. In principle the method could be extended to an abitrary photon number N, however the needed computing resources would grow exponentially with N. Furthermore, the correctness of the method was (partially) confirmed by comparing the results to Lee, et al. (Ref. [4]) and finally, it is shown that the difference up to 200% in Fisher infomation is possible, if the specific information of the photon loss is known prior to the measurement. What is important to mention is that the optimisation method discussed in Chapter II, is approximate due to the long time needed for computation. Therefore, its precision, in principle, is not expected to increase if the number of photons N is increased. In cases of higher N’s, global optimisation algorithms should be implemented or the density of sampling of B increased accordingly. Nevertheless, since in any realistic situation N is low, this method gives a correct assessment of the sensitivity limit. Last, but not least, concerning the scope of this work and its contribution to science, the limits to the phase sensitivity were found with a strong focus on practcally realisable systems. Still, the experimental cost of “producing” the quantum states on demand and non-ideal detection schemes existing remain most important technological challanges for the fututre. Therefore, as a forecast, accounting for the effect of these two problems in the mathematical model, would probably be the next step in the current analysis that should bring us closer to the truth. 26
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