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Quantization of free radiation
Semi-classical models of atom-radiation interaction can handle many physi-cal processes, including absorption and stimulated emission in lasers, where matter is treated as quantum while light is a classical eld . However, some other phenomena can not be described in this semi-classical model. For instance, spontaneous emission can only be treated correctly using a fully quantum framework, where both the radiation and the matter are quan-tized . Furthermore, it is found that a free electromagnetic eld, far from sources, shows a series of new properties  such as interference of photons   , squeezed light , quantum uctuations of vacuum, etc., which can not be described by only the classical elds. This chapter will introduce the quantization of free electromagnetic elds and its associated quantum properties of light.
Annihilation and creation operators
It is well known how to quantize a system of material particles in quantum mechanics. The classical problem rstly is written in Hamilton’s canoni-cal form, which expresses the system energy as a function of the particle position ~x and the conjugate canonical momentum p~. Then the classical Hamiltonian H (x1; :::; xi; p1; :::; pi) is replaced by the quantum operators as H (^x1; :::; x^i; p^1; :::; p^i), which obeys the canonical commutation relation : [^xi; p^j] = ih ij; (2.1) where, ij = 1 if i = j, and ij = 0 if i 6= j. We can see that the operators of position x^i and momentum p^i commute with di erent labels of i; j, and don’t commute with the same label i = j; the labels are associated to di erent par-ticles. Importantly, being di erent from classical physics, the commutation relation introduces Heisenberg inequality in quantum mechanics.
To describe quantum properties of light we need introduce the quantiza-tion of light eld, where the light eld is expressed by quantum operators instead of classical electric eld. Similarly, the procedures to quantize free radiation (the free electromagnetic eld) are :
I. Get the solutions of Maxwell’s equation in the basis of plane waves, which are a set of electric elds with di erent frequencies and polarizations;
II. Write the solutions with the electric eld normal variables, normalized to photon numbers;
III. Express the energy (Hamiltonian) in a form of harmonic models; IV. Introduce the annihilation and creation operators of quantum harmonic oscillators and get quantized Hamiltonian of free radiation, so the commutation relation is introduced in the conjugate canonical variables of radiation elds.
Using the solutions of Maxwell’s equations, the analytic signal of electrical eld, in a quantization volume with a size L, can be expressed of a sum of orthogonal plane waves with discrete frequencies1, E(+) (~r; t) = i El lei kl ~r !lt : (2.2)
In the expression above, El = h!l=2 0L3 is the normalization factor related to the energy of classical elds in the mode l; l is the complex normal variable of mode l associated to photon numbers.
Thus, the radiation energy is given in a sum of the energies related to each normal mode l.
Let us introduce harmonic quantum operators of light also called anni-hilation and creation operators, which are associated to the normal variable l. The time-independent operators a^l satisfy the relation of commutation: [^al; a^ly0] = l;l0 and [^al; a^l0] = 0: (2.4) a^l; a^yl are annihilation and creation operators of photons in the mode l, also called boson operators in quantum mechanics. Replacing the amplitude l by the annihilation operator in Equ. 2.2, the quantization of the light eld, E^(+) r;~ t = i Xl ~lElei(~kl ~r !lt)a^l (2.5)
The eigenstates of the annihilation operators are coherent states or quasi-classical states, which in a mode l can be expressed as below , a^lj li = lj li; (2.8) where, l is the eigenvalue, which is the complex amplitude of the eld. Coherent states are the quantum states of classical light sources, for instance, the output of lasers. It is important to note that the annihilation operator a^ is non-Hermitian, and it is easy to prove that when time t = 0 any coherent state j li with a complex amplitude l is a superposition of Fock states, j li = e p jnli: (2.9) =0 nl!
Up to now we have quantized free radiation. Compared to classical elds, we introduced annihilation and creation quantum operators, and the corre-sponding commutation relations, which therefore can further describe quan-tum properties of non-classical states of light.
For the quantization of elds we have narrow band approximation. When ! !0 !0, E0 = h!0=2 0L3 which is not photon frequency ! dependent. We can factorize E0 in the quantization of the eld, ^(+) X ~ E (~r; t) = iE0 ei(kl ~r !lt)a^l: (2.10)
In the above equation, the plane wave ei(kl ~r !lt) is the classical mode, and a^l is the annihilation operator related to this mode.
The phasor representation and Heisenberg in-equality
The annihilation and creation operators a^, a^y are non-Hermitian, therefore non-observable. Thus in practical measurement, we de ne observable Her-mitian operators using time-independent a^, a^y. The quadrature operators are formed as below, x^l = a^ly + a^l and p^l = i a^ly a^l ; (2.11)
where x^l and p^l are called amplitude and phase quadrature operators re-spectively, which are either time-independent.2 Classically, the quadrature components correspond to real and imaginary parts of analytic signal in Equ. 2.2. A more general way, the quadrature operator can be de ned with the angle as below, x^l = ei a^ly + e i a^l: (2.12)
As seen in Fig. 2.1, a complex light eld, which corresponds to a coherent state, is represented with a vector arrow in phasor representation, where amplitude and phase quadrature components correspond to two axis of the coordinate3. The quadrature operators are Hermitian, and their eigenvalues indicate the classical quadratures of the elds. We can detect quadratures of a light eld directly via homodyne measurements, which will be introduced in chapter 4.
According to the commutation relation between annihilation and creation operators, as in Equ. 2.4, we nd the commutation relation between conju-gate quadrature operators, [^xl; p^l0] = 2i l;l0 and [^xl; x^l0] = [^pl; p^l0] = 0: (2.13)
Thus, we can get Heisenbery inequality, xl2 pl2 1; (2.14) where x2 = h( x^)2i and x^ = x^ h x^i represent the uctuations of the operator x^. The coherent states saturate the Heisenberg inequality in Equ. 2.14, and satisfy, xl2 = pl2 = 1; (2.15) where the unit is an energy related term [El]2, de ned in Equ. 2.2.
In intensity or phase measurements of light, the sensitivity always has a standard quantum limit, also called shot noise limit, which originates from the Heisenberg inequality of coherent states. Because the coherent states, and the coherent vacuum which has a zero mean eld, have the same Heisenberg inequality 1, we call quantum uctuation of coherent light vacuum uctua-tion.
Here, for both amplitude and phase quadrature operators, the Heisenberg inequality is because of vacuum uctuations of free radiation, which can not be described by a classical way. We can think the quantum description of free radiation corresponds to a classical eld plus vacuum uctuations, as seen in Fig. 2.1, where the red circle represent vacuum uctuations.
Usually in lasers, the process of photon generation is random, and this random process gives rise to shot noise limit in intensity and phase measurements of laser light. For such coherent states, the photon number follows p a Poisson distribution with a standard deviation of N, which is a statis-tic property of classical radiation; sub-Poisson distribution corresponds to non-classical source, which is anti-bunch in the photon generation process.
We have seen that the coherent states have the same vacuum uctuations in both amplitude and phase quadratures, as seen in Equ. 2.15. The sensitivity of intensity and phase measurements with lasers will be limited by the shot noises, which origin from random generation processes of photons.
Here we will present a type of quantum resources, called squeezed light . Interestingly, this type of light can exceed the standard quantum limit. As seen in Fig. 2.2, it is shown that the phase representation of squeezed states, where the uctuation of one quadrature exceeds the quantum limit 1, and the uctuation of the corresponding orthogonal quadrature is bigger than 1, conserving Heisenbery inequality. This means that potentially, we can increase measurement sensitivity by using squeezed light because of less quantum uctuations .
The squeezing operator Ar, where r is the real squeezing coe cient, is de ned as below, Figure 2.2: Phasor representation of a squeezed state. The vector arrow corresponds to the eld vector, and the red circle represents the quantum uctuations, which obeys Heisenberg relation. Di erent from coherent states in Fig. 2.1, the variance of the amplitude quadrature is less than 1; the phase part is bigger than 1.
From above equations, we see that with a squeezing operation K acting on coherent states, we can get a new state with squeezed variance e 2r in a quadrature and anti-squeezed variance e2r in the orthogonal quadrature.
Squeezed quadratures have less quantum uctuations, so the sensitivity of measurements on the corresponding squeezed quadrature can be improved beyond shot noise limit. In metrology, the sensitivity of measurements can exceed the standard quantum limit via using squeezing on the corresponding quadrature  .
Figure 2.3: A spectral mode basis. The spectrum (envelope) of a frequency comb is divided into six bands, corresponding to a train 100 fs pulses centered at 795nm with a bandwidth of 10 nm in the time domain.
Modes of light elds
The quantization of free radiation in the previous chapter described elec-tric eld as a sum of harmonic oscillators, and the Hamiltonian is expressed as in Equ. 2.6. In this case, the harmonic oscillators correspond to the plane waves with the frequency !l = c k ~ k. These plane wave modes construct a basis carrying the energies and information, which are photon numbers in quantum description, and are the amplitude classically. The modes are classical conceptions originally from Maxwell’s equations. How-ever, in practice, many other kind of modes of light are also often used, such as Hermit-Gaussian spatial modes, temporal modes of ultrafast pulses, fre-quency modes, polarizations of light, and output modes of light resources4. In our work, we divide a 10 nm bandwidth spectrum of optical frequency combs into many frequency pixels (spectral bands), and these spectral bands construct a measurement basis, as seen in Fig. 2.3.
A set of orthogonal modes fui (~r; t)g of light, where z is the axis of prop-agation direction, ~r = (~; z) and ~ is the transverse coordinate, constructs a mode basis, if it satis es the condition as below, Zt0t0+T ZS ui (~r; t) uj (~r; t) d~rdt = ij; (2.20) where S is the surface of detection and T is the measurement time.
In this mode basis, the electric eld operator can be expressed where A is the normalization factor, which we will often omit in the rest of the manuscript. Here we express the eld operator with sum of a set of orthogonal modes, which we will introduce in detail in next section.
In Equ. 2.21, di erent modes can construct a di erent basis, and the same eld can be expressed in di erent basis. Let us consider there is another basis consist of a set of modes fwi(~r; t)g, which equals a linear transform U acting on the basis ~u,
In general, with a basis change, a state of light can be represented in di erent basis; also a basis change can be any unitary transform, usually realized by linear optical networks . In our experiment, S is big enough compared to the beam transverse size, and the spatial mode is only TEM00, so the integral of S can be neglected, and only frequency and temporal modes are considered.
Here Uij is also a projection operation, where we project the modes fuig onto another mode basis fwjg. As the unitary transform corresponds to a square matrix, only when the two mode basis have the same number of dimensions, U is a unitary transform, or basis change; if not, it is only pro-jection with the form of Equ. 2.22, but then it is not unitary.
The physical interpretation of a basis change is corresponding a unitary transform of the eld, where the energy and commutation relations before and after transform are conserved. In practice, this can be implemented with linear optics without loss, such as optical beam splitter, phasors.
Monomode and multimode
In our work, the important property is that our quantum resource is multimode, not monomode, therefore we give the de nition as following.
De nition: a pure state j i is monomode if it exists a basis of modes fvig, in which we have, j i = j i j0; :::; 0; :::i; (2.28) where j i which is not vacuum is the state in the rst mode of the basis fvi g, and the states of all the other modes are vacuum j0i. A quantum state is multimode if it is not monomode. Here we give an example of multimode squeezing, for example, it can be generated by 4 independent OPOs, j i = j0sqz1i j0sqz2i j0sqz3i j0sqz4i j0; :::; 0; :::i; (2.29) where the leading four modes are independently squeezed by four individual OPOs, and the other modes are vacuum, which can not be reduced into the form of Equ. 2.28.
Importantly, quantum correlation, for example, two-partite entanglement  can be generated via mixing monomode squeezing and vacuum by optical beam splitter as seen in Fig. 2.4, but this kind of entangled state can be reduced to the form of monomode as in Equ. 2.28. For instance, in the input basis, the state is expressed with only one squeezed mode and vacuum,
Representation in the continuous variable regime
Previously, we de ned quadrature operators in Equ. 2.11, which are contin-uous variables. In this chapter we will introduce how to represent a gaussian state in continuous variable regime. First we simply give the representa-tion with density matrices, and also in continuous variable case with Wigner function. Then, in particular for gaussian case, the covariance matrix will be presented.
Gaussian states and covariance matrices
Quantum states are gaussian states, if their corresponding Wigner function has a gaussian shape, otherwise it is called non-gaussian states. In the gaus-sian case, the Wigner function is only determined by the rst two moments of the quadratures. The rst is the mean values of the quadratures in di erent corresponding modes; the second is the covariances of the quadratures in dif-ferent modes, which originate from the uctuations of the quadratures, such as h( x^)2i, h( p^)2i, h p^ x^ i and h x^ p^i. Hence, all the uctuation property of gaussian states can be expressed with the covariance matrix. A quantum state, in the basis with N modes, can be written in a covariance matrix V .
The full covariance matrix is composed of four parts of correlations. The diagonal is amplitude quadrature correlation h( x^)2iji, and phase quadrature correlation h( p^)2iji; the o -diagonal is amplitude-phase correlation h x^i p^ji, and phase-amplitude correlation h p^i x^ji. Each of the four parts is a real and symmetry matrix, and most of quantum resources have only amplitude and phase quadrature parts, without o -diagonal parts, such as, SPOPO , cascading four wave mixing , ect. Therefore linear optical networks are applied to obtain amplitude-phase or phase-amplitude correlations, for instance, the implementation of controller-z gates with o -line OPOs  .
The covariance matrix is often measured via balanced homodyne detec-tion, which will be presented later. Notice that, the covariance originates from both classical and quantum uctuations , and when there is no clas-sical noise (shot noise limited), the covariance matrix represents the quantum uctuations and correlations.
Table of contents :
I Theory and Experiment of the SPOPO
1 Optical frequency combs
1.1 Classical electric elds
1.2 Optical frequency combs
1.2.1 Denitions of frequency combs and ultrafast pulses
1.2.2 Frequency dependent phase
1.3 Dispersion of linear media
1.4 Synchronized optical cavity
2 Quantum optics in continuous variable regime
2.1 Quantization of free radiation
2.1.1 Annihilation and creation operators
2.1.2 The phasor representation and Heisenberg inequality
2.1.3 Squeezed states
2.2 Modes of light elds
2.2.1 Basis change
2.2.2 Monomode and multimode
2.2.3 Basis change in the quadrature representation
2.3 Representation in the continuous variable regime
2.3.1 Density matrix and Wigner function
2.3.2 Gaussian states and covariance matrices
2.3.3 Basis change of covariance matrix
2.4 Symplectic transform
2.5 Williamson decomposition and Bloch Messiah reduction
2.5.1 Recipe of Williamson decomposition
3 SPOPO model and simulation
3.1 Basic tools of nonlinear optics
3.1.1 Propagation equation of nonlinear optics
3.2 Nonlinear eect with ultrafast pulses
3.2.1 Second order polarization
3.2.2 The wave number of ultrafast pulses
3.3 The SPOPO model
3.3.1 Preparation of pump, frequency doubling
3.3.2 Parametric down conversion with optical frequency combs
3.3.3 Schmidt modes
3.3.4 Squeezing of SPOPO
3.4 Simulating multimode correlation of the SPOPO
3.4.1 Simulating covariance matrix
3.4.2 The eigenmodes
4 Principle of the SPOPO experiment and preparation of light source
4.1 Principle of the experiment
4.1.1 The objective
4.1.2 Experimental conguration
4.2 Laser Source
4.2.1 Pump laser
4.2.2 Femtosecond laser
4.2.3 Pump pointing locking
4.2.4 Spectral locking
4.3 Preparation of laser
4.3.1 Correction of the astigmatism
4.3.2 Dispersion compensation
4.3.3 Frequency doubling
4.3.4 Mode matching and relative delay
4.4 SPOPO cavity
4.4.1 Cavity conguration
4.4.2 Alignment of the cavity
4.4.3 PDH locking
4.4.4 Alignment of the nonlinear crystal and the pump
4.4.5 Amplication and deamplication
4.4.6 The threshold of the SPOPO
4.4.7 Relative phase locking between signal and pump
4.4.8 The SPOPO above threshold
4.5 Homodyne detection with pulse shaping
4.5.1 Homodyne detection
4.5.2 Pulse shaping in the local oscillator
4.5.3 Measurements and data collecting
4.6 State reconstruction with 16-pixel covariance matrix
4.6.1 Measuring multimode quantum noises of SPOPO and covariance matrix
4.6.2 Multimode analysis
4.6.3 Full multipartite entanglement
II Quantum Network
5 Continuous-variable cluster states
5.1 Cluster states in CV
5.2 Cluster states with squeezed states and linear optics
5.3 Generating cluster states with OPOs
6 Simulating quantum networks via pulse shaping
6.1 EPR network via pulse shaping
6.2 Simulating quantum networks with SPOPO
6.3 Witness of quantum networks
6.4 Simulating a multipartite quantum secret sharing
6.6 Appendix 1: Optimization of cluster matrix
6.7 Appendix 2: Reconstructed secret modes
III Multipixel-homodyne based Quantum Computing and Metrology
7 Quantum computing with SPOPO
7.1 Multipixel homodyne detection
7.2 Setup and alignment
7.3 Simultaneously measuring multimode covariance matrix
7.4 Characterization of Feasible operations
7.5 Feasible quantum networks
7.6 Locking squeezed vacuum
8 Multimode entanglement with cascading FWM
9 Quantum frequency metrology
9.1 Mode-dependent characters of ultrafast pulses
9.2 Setup and frequency-resolved measurement
9.3 A quantum spectrometer
9.3.2 Experimental conguration
9.3.3 Central frequency measurement
9.3.4 Multimode analysis of frequent metrology