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From the Weyl quantization of a particle on the circle to number–phase Wigner functions
Annals of Physics 351 (2014) 919–934
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From the Weyl quantization of a particle on the circle to number–phase Wigner functions Maciej Przanowski, Przemysław Brzykcy, Jaromir Tosiek ∗ Institute of Physics, Technical University of Łódź, Wólczańska 219, 90-924 Łódź, Poland
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Article history: Received 14 April 2014 Accepted 18 October 2014 Available online 26 October 2014 keywords: Quantum phase Number–phase Wigner function
abstract A generalized Weyl quantization formalism for a particle on the circle is shown to supply an effective method for defining the number–phase Wigner function in quantum optics. A Wigner function for the state ϱˆ and the kernel K for a particle on the circle is defined and its properties are analysed. Then it is shown how this Wigner function can be easily modified to give the number–phase Wigner function in quantum optics. Some examples of such number–phase Wigner functions are considered. © 2014 Elsevier Inc. All rights reserved.
1. Introduction The present paper is a continuation of our previous work [1]. In fact, it has been motivated by a question raised by the referee of [1]. Namely, the referee pointed out that the Weyl quantization formalism developed in [1] should be closely related to the Wigner function and the Wigner representation of quantum phase investigated previously by the others [2–4]. So, here we follow this suggestion and we intend to display, how one can define the Wigner function which depends on the number and the phase. We arrive at this goal by employing the generalized Weyl quantization formalism for a particle on the circle given in [1]. As will be shown the answer to this question can be easily found and, moreover, it appears to be fairly natural within the generalized Weyl quantization machinery. It is well known that the Wigner function for a system of particles in R3 is a real function on the corresponding classical phase space so it depends on the Cartesian coordinates of the particles
∗
Corresponding author. E-mail addresses: [email protected] (M. Przanowski), [email protected] (P. Brzykcy), [email protected] (J. Tosiek). http://dx.doi.org/10.1016/j.aop.2014.10.011 0003-4916/© 2014 Elsevier Inc. All rights reserved.
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and on the respective canonically conjugate momenta [5–7]. The Wigner function is uniquely defined by the density operator of the system and, conversely, the density operator is uniquely determined by the given Wigner function. Then, the expectation value of any quantum observable in a given quantum state can be found by integrating the product of the corresponding classical observable and the respective Wigner function. This procedure resembles very much the one well known in statistical physics. The only important difference consists in the fact that, in general, the Wigner function is not pointwise non-negative. Consequently, in general, it does not represent a probability distribution but, as is usually said, the Wigner function is a quasi-probability distribution. Nevertheless, the marginal distributions of any Wigner function give the probability distributions for coordinates and momenta, respectively. The natural question arises if one can define the analogous Wigner functions for the constraint quantum systems (e.g. a particle on the circle) or for the quantum systems described by the finite-dimensional Hilbert spaces. This question has raised a great deal of interest and many authors have analysed various examples of the Wigner functions for such quantum systems [8–19] (in particular see [18] and the references therein). Another interesting question, which turns out to be closely related to the previous one, concerns the definition of the Wigner function in quantum optics as a function of the photon number and the phase. As is known, the photon number and the phase can be considered as the canonically conjugated quantities and therefore, one attempts to find the Wigner function that depends on these quantities i.e. the so called number–phase Wigner function. This problem was explored by numerous authors. For example, J.A. Vaccaro and D.T. Pegg [20] defined a number–phase Wigner function employing the results of W.K. Wootters on the discrete Wigner functions in finite-dimensional spaces [13]. Such an approach is undoubtedly based on the celebrated Pegg–Barnett formalism in the theory of quantum phase [21–23]. A. Lukš with V. Peřinová [2], and M.R. Hush et al. [4] defined a number–phase Wigner function which was extended to rather unphysical half integer values of the number of photons. In the distinguished work by J.A. Vaccaro [3] a number–phase Wigner function is proposed assuming at the very beginning that this function should satisfy, besides the usual properties inflicted on Wigner function [7], some additional property leading to the interference fringes for the Wigner functions of the Schrödinger cat states. The aim of our paper is to show that the number–phase Wigner function can be easily defined with the use of the generalized Weyl quantization formalism on the cylindrical phase space S 1 × R1 under the observation that Hilbert space L2 (S 1 ) can be considered as an enlarged Fock space HF . The paper is organized as follows. In Section 2 we recall the main formulas of the generalized Weyl quantization formalism for the cylindrical phase space S 1 × R1 . The generalized Stratonovich–Weyl quantizer for an arbitrary kernel K is introduced and its properties are studied. The definition of the Wigner function (for an arbitrary kernel K ) in S 1 × R1 is given in Section 3. Main properties of the Wigner functions are also analysed. In Section 4 the Wigner function is specified to the case when the kernel K = KS leads to the symmetric ordering of operators under the Weyl quantization prescription. The number–phase Wigner function is defined in Section 5. We show there that having defined the Wigner function in S 1 × R1 for the kernel K = KS and employing the results of our previous work on quantum phase [1] one can quickly define a number–phase Wigner function in quantum optics. The properties of this Wigner functions are also studied in Section 5. Section 6 is devoted to some explicit examples of the number–phase Wigner functions. We consider the Fock states, the coherent states, the squeezed states, the black body radiation and the ‘Fock cat’ states. Finally, concluding remarks in Section 7 end the paper.
2. The generalized Weyl quantization on the cylinder Consider a particle on the circle S 1 . Let Θ ∈ [−π , π ) denote the angle coordinate of the particle (Important remark: In our paper the interval [−π , π ) is identified with the circle S 1 ) and L ∈ R1 be the angular momentum of this particle. The corresponding phase space is the cylinder S 1 × R1 . Given a function f = f (Θ , L) on S 1 × R1 one assigns to it an operator W [K ](f ) in the Hilbert space L2 (S 1 ) according to the rule [24,1] f = f (Θ , L) → W [K ](f )
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
W [K ](f ) :=
∞
π
K (σ , l)
−π
l=−∞
∞
n=−∞
π
921
f (Θ , n})
−π
dΘ dσ × exp −i(σ n + lΘ ) Uˆ (σ , l) 2π 2π ∞ π dΘ ˆ [K ](Θ , n) = f (Θ , n})Ω , 2π n=−∞ −π
(2.1)
where the function K = K (σ , l), σ ∈ [−π , π ), l ∈ Z is called the kernel and Uˆ (σ , l) is the unitary operator on the Hilbert space L2 (S 1 )
σ
Uˆ (σ , l) = exp i
ˆ Lˆ + lΘ
} i i ˆ exp σ l exp ilΘ σ Lˆ = exp 2 } i i ˆ = exp − σ l exp σ Lˆ exp ilΘ 2
=
∞
}
exp iσ
k+
k=−∞
l 2
|k + l⟩⟨k|,
σ ∈ [−π , π ).
(2.2)
The quantization rule (2.1) is called the generalized Weyl quantization with the kernel K . Moreover,
ˆ [K ](Θ , n) := Ω
π
∞ l=−∞
K (σ , l) exp −i σ n + lΘ
Uˆ (σ , l)
−π
dσ 2π
(2.3)
is the generalized Stratonovich–Weyl (GSW) quantizer for the kernel K . Recall that Uˆ (σ , l) has the following properties [17,25]
Tr Uˆ (σ , l) = 2π δl0 δ (S ) (σ ),
(2.4a)
Tr Uˆ Ď (σ , l)Uˆ (σ ′ , l′ ) = 2π δll′ δ (S ) (σ − σ ′ ),
(2.4b)
where δ (S ) (σ ) stands for the Dirac delta on the circle given by
δ (S ) (σ ) =
1
∞
2π l=−∞
exp ilσ .
(2.5)
As can be easily shown (see for example [6,1]), the natural assumptions about quantization impose some restrictions on the kernel K . Namely
ˆ ) for an arbitrary function f depending only on Θ , f = f (Θ ) iff (i) W [K ](f ) = f (Θ ∀l∈Z K (0, l) = 1,
(2.6a)
(ii) W [K ](f ) = f (Lˆ ) for an arbitrary function f depending only on L, f = f (L) iff
∀σ ∈[−π,π) K (σ , 0) = 1,
(2.6b)
(iii) the operator W [K ] is symmetric for any real function f = f (Θ , L) iff
∀σ ∈[−π,π), l∈Z K ∗ (σ , l) = K (−σ , −l) (where the star ‘∗’ stands for the complex conjugation).
(2.6c)
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Then the GSW quantizer (2.3) has the following important properties. (a) If (2.6a) or (2.6b) holds true then
ˆ [K ](Θ , n) = 1. Tr Ω
(2.7a)
(b) The condition (2.6c) yields
ˆ Ď [K (Θ , n)] = Ω ˆ [K (Θ , n)]. Ω
(2.7b)
(c)
ˆ [K ](Θ , n)Ω ˆ [K ](Θ ′ , n′ ) Tr Ω
=
1
∞
π
2π l=−∞ −π
K (σ , l)K (−σ , −l)
× exp i[σ (n − n′ ) + l(Θ − Θ ′ )] dσ .
(2.7c)
(d) If (2.6a) holds true then ∞
1
ˆ [K ](Θ , n) = |Θ ⟩⟨Θ |. Ω
(2.7d)
ˆ [K ](Θ , n)dΘ = |n⟩⟨n|. Ω
(2.7e)
2π n=−∞ (e) If (2.6b) holds true then
1 2π
π −π
Let fˆ be an operator in L2 (S 1 ) which arises from some function f (Θ , n}) on the quantized cylindrical phase space S 1 × Z by the generalized Weyl quantization prescription (2.1) and let gˆ be any operator in L2 (S 1 ). Then from (2.1) one quickly gets
Tr fˆ gˆ = Tr
∞
n=−∞
=
π
∞ n=−∞
π
ˆ [K ](Θ , n) f (Θ , n})Ω
−π
f (Θ , n})
−π
1 2π
dΘ
2π
gˆ
ˆ [K ](Θ , n)ˆg dΘ . Tr Ω
(2.8)
Two important remarks are to be made here. Remark 1. For K = 1 our generalized Weyl quantization rule (2.1) is equivalent to the one introduced by J.P. Bizarro in his distinguished paper [16] (see subsection C of the section III of [16]) and also to that considered by J.F. Plebański et al. in [17]. Then the generalized Weyl quantization rule (2.1) in a slightly different notation was given also by Rigas et al. [19] (see Eqs. (40) and (43) in [19]). However, one quickly realizes that the kernel K considered in [19] is necessarily of the form
K (σ , l) = exp i α(l, −σ ) −
1 2
σl ,
α∗ = α
(2.9)
which excludes the case of K (σ , l) = cos σ2l investigated in the following sections of the present work. Remark 2. The generalized Weyl quantization rule (2.1) with the kernel K satisfying (2.6a) for f (Θ , L) = Θ leads to the self-adjoint angle operator [26,1]
ˆ =i Θ
∞ (−1)j−k |j⟩⟨k|. j−k j,k=−∞
(2.10)
j̸=k
ˆ and sin Θ ˆ are self-adjoint operators which correspond to functions cos Θ Then the operators cos Θ ˆ , sin Θ ˆ and sin Θ , respectively. Moreover, the operators Lˆ , cos Θ
constitute the generators of the
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Euclidean group E (2) and they play a crucial role in quantization on the cylinder proposed by K. Kowalski and J. Rembieliński [27] and H.A. Kastrup [28]. In our approach, which is founded on the generalized Weyl quantization prescription (2.1), the canonically conjugate variables are the angle ˆ respectively. We ˆ and L, Θ and the angular momentum L, which are represented by the operators Θ suppose that from the ‘point of view of a particle’ on the circle the angle Θ is more natural variable than cos Θ and sin Θ (or equivalently eiΘ ). 3. Wigner function Let ϱˆ be a density operator of a particle on the circle. It satisfies the usual conditions
ϱˆ + = ϱ, ˆ
(3.1a)
⟨ψ|ϱ|ψ⟩ ˆ ≥ 0 ∀|ψ⟩ ∈ L (S ), Tr ϱˆ = 1. 2
1
(3.1b) (3.1c)
For any observable represented by the operator fˆ the expectation value of this observable in the state ϱˆ is given by the well known formula
⟨fˆ ⟩ = Tr fˆ ϱˆ .
(3.2)
Assume that the operator fˆ arises from some classical observable f = f (Θ , L) as the result of the generalized Weyl quantization rule (2.1). So by (2.8) the relation (3.2) can be rewritten in the form
⟨fˆ ⟩ =
∞ n=−∞
π
f (Θ , n})
−π
1 2π
ˆ [K ](Θ , n)ϱˆ dΘ . Tr Ω
(3.3)
Introducing the function (see also [19])
ϱW [K ](Θ , n}) :=
1 2π
ˆ [K ](Θ , n)ϱˆ Tr Ω
(3.4)
which will be called the Wigner function for the state ϱˆ and the kernel K one writes (3.3) as
⟨fˆ ⟩ =
∞ n=−∞
π
f (Θ , n})ϱW [K ](Θ , n})dΘ .
(3.5)
−π
Observe that for K = 1 one gets the Wigner function introduced by M. Berry [8] and N. Mukunda [9,10] and then studied in all details by J.P. Bizarro [16]. The same function appears in [17,19,29]. Eq. (3.5) resembles very closely the fundamental formula from classical statistical mechanics defining the expectation value of the observable f = f (Θ , n}). Therefore we can identify ⟨fˆ ⟩ ≡ ⟨f (Θ , n})⟩. One easily finds that if the condition (2.6c) is fulfilled then by (2.7b) and (3.1a) we have ∗ ϱW [K ](Θ , n}) = ϱW [K ](Θ , n})
(3.6)
i.e. ϱW [K ] is a real function. Assume that (2.6a) holds true. Performing summation over n of both sides of (3.4) and employing (2.7d) one gets ∞
ϱW [K ](Θ , n}) = Tr |Θ ⟩⟨Θ |ϱˆ = ⟨Θ |ϱ| ˆ Θ ⟩ =: P (Θ ).
(3.7)
n=−∞
The function P (Θ ) given by (3.7) is the probability distribution of the angle Θ in the state ϱˆ . Analogously, assuming (2.6b), performing integration with respect to Θ of both sides of (3.4) and, finally employing (2.7e) one has
π
−π
ϱW [K ](Θ , n})dΘ = Tr |n⟩⟨n|ϱˆ = ⟨n|ϱ| ˆ n⟩ =: P (n})
(3.8)
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i.e. the probability distribution of the angular momentum L in the state ϱˆ . Note that any Wigner function should have the properties described by (3.6)–(3.8) [3,7,16]. It is worthwhile noticing that the discrete Wigner functions have been also extensively studied from the general point of view [29–34]. Moreover, a class of discrete-time Wigner functions has been introduced in signal processing [35], where Wigner distributions have been used for time–frequency analysis [36]. 4. Symmetric ordering of operators Wigner functions of a particle on the circle for K = 1 (the Weyl ordering) have been considered in [8–10,12,16,17,19,29,32,34]. In particular, recently [19] the Wigner functions corresponding to the coherent states on the circle have been studied. It should be noted that in [19] the definition of Wigner function (3.4) with the kernel K of the form (2.9) was given (in different notation) but further analysis has been carried out for K = 1. However, as shown in the next section, if one intends to carry over the results on the Wigner functions in the cylindrical phase space S 1 × R1 to the case of number–phase Wigner functions in quantum optics, it is more convenient to deal with the kernel which used in (2.1) leads to the symmetric ordering of operators. As it has been shown in [24,37–39,1] such a kernel reads
K (σ , l) = cos
σl
2
≡ KS (σ , l)
(4.1)
and it fulfils all the conditions (2.6a)–(2.6c). Therefore the relations (2.7a)–(2.7e) are also satisfied. Moreover, the respective Wigner function ϱW [KS ] has the basic properties (3.6)–(3.8). Substituting (4.1) into (2.3) one quickly gets
ˆ [KS ](Θ , n) = Ω
∞ 1
2 k=−∞
exp [−i(n − k)Θ ] |n⟩⟨k| + exp [i(n − k)Θ ] |k⟩⟨n|
= π |n⟩⟨n|Θ ⟩⟨Θ | + |Θ ⟩⟨Θ |n⟩⟨n| ,
(4.2)
where ∞
1
|Θ ⟩ = √
2π k=−∞
exp (−ikΘ ) |k⟩
(4.3)
ˆ and ⟨n|Θ ⟩ = √1 exp (−inΘ ) (for detailed is the normalized eigenvector of the angle operator Θ 2π
ˆ see for instance [26,1]). Inserting (4.2) into (3.4) we have analysis of the angle operator Θ ϱW [KS ](Θ , n}) =
1 2π
∞
Re
exp [−i(n − k)Θ ] ⟨k|ϱ| ˆ n⟩
k=−∞
= Re ⟨Θ |ϱ| ˆ n⟩⟨n|Θ ⟩ .
(4.4)
The function ϱW [KS ](Θ , n}) given by (4.4) defines the density operator ϱˆ uniquely. Indeed, one can rewrite (4.4) in the following form
ϱW [KS ](Θ , n}) =
1
∞
2π k=−∞
cos[(n − k)Θ ]Re⟨k|ϱ| ˆ n⟩ + sin[(n − k)Θ ]Im⟨k|ϱ| ˆ n⟩ .
(4.5)
From (4.5) we easily get Re⟨k|ϱ| ˆ n⟩ = 2
π
ϱW [KS ](Θ , n}) cos[(n − k)Θ ]dΘ for k ̸= n,
(4.6a)
−π
⟨n|ϱ| ˆ n⟩ =
π
ϱW [KS ](Θ , n})dΘ ,
(4.6b)
−π
Im⟨k|ϱ| ˆ n⟩ = 2
π −π
ϱW [KS ](Θ , n}) sin[(n − k)Θ ]dΘ
(4.6c)
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
925
√ (of course (4.6b) follows also from (3.8)). Multiplying both sides of (4.6c) by i = −1 and adding to (4.6a) one finally has
⟨k|ϱ| ˆ n⟩ = 2 ⟨n|ϱ| ˆ n⟩ =
π
ϱW [KS ](Θ , n}) exp[i(n − k)Θ ]dΘ for k ̸= n,
−π π
ϱW [KS ](Θ , n})dΘ .
(4.7)
−π
Concluding, the formulas (4.7) give all matrix elements of ϱˆ in the angular momentum basis {|n⟩}∞ ˆ itself. n=−∞ , ipso facto the density operator ϱ 5. Number–phase Wigner function The problem of defining the quantum phase of a harmonic oscillator or of a single-mode electromagnetic field has a long and involved history initiated by P.A.M. Dirac [40] and F. London [41] in the years 1926–27. We are not going to discuss all meanders of that history and we refer the reader to some of numerous works devoted to this question [21–23,42–58,1]. In the present paper we employ the results of our recent work [1], where it is argued that a reasonable way to define the quantum phase consists in extending the Fock space of the harmonic oscillator or the single-mode electromagnetic field to the Hilbert space L2 (S 1 ). This idea has been considered by several authors [49–53,56,58]. The construction given in Ref. [1] can ∞be stated as follows. Let HF be the respective Fock space and denote the Fock basis of HF by |n⟩ n=0 . (Remark: The vectors of HF will be marked
by the additional under-bar i.e. |n⟩, |ψ⟩, |χ⟩, . . . etc.) We embed HF in the Hilbert space L2 (S 1 ) by Jˆ : HF ∋
∞
cn |n⟩ →
n =0
∞
cn |n⟩ ∈ L2 (S 1 ),
cn ∈ C.
(5.1)
n = 0, 1, 2, . . . n = −1, −2, . . .
(5.2)
n =0
ˆ of L2 (S 1 ) onto HF by Define the projection Π 2 1 ˆ : L2 (S 1 ) ∋ |n⟩ → |n⟩ ∈ HF , Π L (S ) ∋ |n⟩ → |θ⟩ ∈ HF ,
where |θ ⟩ is the null vector in the space HF and put
Ď ˆ := Π ˆ |ψ⟩ ⟨ψ|Π Ď ⟨χ|Jˆ := Jˆ|χ⟩ .
(5.3a) (5.3b)
Then to any classical observable being a function of the phase φ , f = f (φ) one assigns the corresponding quantum observable in a state |ψ⟩ ∈ HF by quantizing the classical observable f (−Θ ) on the circle in the state Jˆ|ψ⟩ ∈ L2 (S 1 ) using the generalized Weyl quantization rule (2.1). The expectation value of the quantum observable corresponding to f = f (φ) in the state |ψ⟩ is equal to the expectation value of f (−Θ ) calculated for Jˆ|ψ⟩ ∈ L2 (S 1 ). [The substitution f (φ) → f (−Θ ) is justified by the commonly
ˆ (|Θ ⟩) = |−Θ ⟩.] As has been used conventions given by (4.3) and (5.15) which lead to the result Π shown in [1], the procedure described above is equivalent to the approach developed by J.H. Shapiro and S.R. Shepard [46], and by P. Bush, M. Grabowski and P.J. Lahti [48], where the quantum phase is given as the positive operator valued (POV) measure on [−π , π ) M0 : B [−π , π ) ∋ X →
∞ 1
2π j,k=0
exp [i(j − k)φ ] dφ X
|j⟩⟨k|
(5.4)
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M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
with B [−π , π ) standing for the family of Borel sets on [−π , π ). This POV measure is a compression ˆ E (X )Π ˆ of the spectral measure E i.e. M0 (X ) = Π E : B [−π , π ) ∋ X →
1
∞
2π j,k=−∞
exp [i(j − k)φ ] dφ
|j⟩⟨k|.
(5.5)
X
ˆ defined by (5.2) is just a Naimark projection [48,59,60]. Moreover, one can also Thus the projection Π show [1] that our procedure is equivalent to the Pegg–Barnett approach [21–23] but without introducing any, rather artificial, finite-dimensional Hilbert spaces. Now we are at the position, when the number–phase Wigner function can be defined. Let ϱˆ =
∞
ϱjk |j⟩⟨k|
(5.6)
j,k=0
be a density operator on the Fock space HF . Then the operator
ϱˆ = Jˆϱˆ Jˆ =
∞
ϱjk |j⟩⟨k|
(5.7)
j,k=0
is the density operator on L2 (S 1 ) associated to ϱˆ . Our idea is to define the number–phase Wigner function corresponding to the state ϱˆ (5.6) on HF by the Wigner function ϱW [K ](Θ , n}) corresponding to the state ϱˆ (5.7) on L2 (S 1 ). More precisely, the number–phase Wigner function for the state ϱˆ and the kernel K is defined as
ϱW [K ](φ, n) := ϱW [K ](−φ, n}) =
1 2π
ˆ [K ](−φ, n)ϱˆ , Tr Ω
n ≥ 0.
(5.8)
This approach based on Eq. (3.3) leads directly from the Stratonovich–Weyl quantizer to the Wigner function. It differs from the approach which starts from some axioms imposed on the Wigner function [3,7,16]. Since in HF , n = 0, 1, 2, . . . , for the sake of further consistency one must assume that the kernel K is such that
ˆ [K ](Θ , n)|k⟩ = 0. ∀Θ ∈[−π,π) , ∀j,k≥0, n<0 ⟨j|Ω
(5.9)
From (2.3) with (2.2) we easily infer that the condition (5.9) is equivalent to the following fundamental condition imposed on K
π
∀j,k≥0, n<0
K (σ , j − k) exp iσ
−π
j+k 2
−n
dσ = 0.
(5.10)
One quickly finds that this condition is not fulfilled for K = 1 (the Weyl ordering) and it is satisfied, for example, for K = KS . Therefore, we restrict ourselves to this latter case. From (5.8) applying (4.4), (5.6) and (5.7) one obtains
ϱW [KS ](φ, n) =
1 2π
Re
∞
ˆ n⟩ exp [i(n − k)φ ] ⟨k|ϱ|
k=0
= Re ⟨Θ = −φ|ϱ| ˆ n⟩⟨n|Θ = −φ⟩ ,
n = 0, 1, 2, . . . .
(5.11)
Let f = f (φ) be a classical observable relevant to the phase. Then employing (3.5), (5.8) and (5.11) we find that the expectation value of the quantum observable corresponding to f (φ) in the state ϱˆ given by (5.6) reads
⟨f (φ)⟩ =
∞ π 1
2π n,k=0 −π
f (φ) exp[i(n − k)φ]ϱkn dφ.
(5.12)
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In particular, if ϱˆ is a pure state
ϱˆ = |ψ⟩⟨ψ|,
⟨ψ|ψ⟩ = 1,
(5.13)
the formula (5.12) can be rewritten in the form ∞ π 1
⟨f (φ)⟩ =
2π n,k=0 −π
f (φ) exp[i(n − k)φ]⟨k|ψ⟩⟨ψ|n⟩dφ.
(5.14)
This result is in agreement with the respective result calculated within the Pegg–Barnett formalism (see Eq. (71) in [1]; note that in that equation in [1] the factor 21π is erroneously missing). Define the vector |φ⟩ in the rigged Hilbert space of HF by ∞
1
|φ⟩ := √
2π n=0
exp(inφ)|n⟩.
(5.15)
One quickly finds the relations
π
|φ⟩dφ⟨φ| = 1ˆ
(5.16)
−π
and
⟨φ|φ ′ ⟩ =
1 (S ) 1 i φ − φ′ δ (φ − φ ′ ) + − cot 2 4π 4π 2
(5.17)
(see [44]). From (5.16) we infer that any |ψ⟩ ∈ HF can be written in the form π
|ψ⟩ =
⟨φ|ψ⟩|φ⟩dφ.
(5.18)
−π
Then Eq. (5.14) can be rewritten as
⟨f (φ)⟩ =
π
−π
2
f (φ) ⟨φ|ψ⟩ dφ.
(5.19)
Consequently, the function ψ(φ) := ⟨φ|ψ⟩ can be considered as the wave function in the phase representation for the state |ψ⟩ and |⟨φ|ψ⟩|2 = |ψ(φ)|2 is the respective phase probability distribution. Thus, the vector |φ⟩ defined by (5.15) is the phase state vector as it is assumed in quantum optics (see e.g. [41,44,54–58]). Finally, in terms of |φ⟩ the formula (5.12) reads
⟨f (φ)⟩ =
π
ˆ dφ f (φ)⟨φ|ϱ|φ⟩
(5.20)
−π
and (5.11) takes the form analogous to (4.4)
ϱW [KS ](φ, n) = Re ⟨φ|ϱ| ˆ n⟩⟨n|φ⟩ ,
n = 0, 1, 2, . . . .
(5.21)
The marginal distributions calculated for the number–phase Wigner function (5.21) give the phase probability distribution ∞
ϱW [KS ](φ, n) = ⟨φ|ϱ|φ⟩ ˆ
n=0
=
1 2π
1 + 2Re
∞ k,n=0
exp[i(n − k)φ]ϱkn
=: P (φ)
(5.22)
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M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
(compare with Eq. (6.86) from [57]) and the photon number probability distribution
π −π
ϱW [KS ](φ, n)dφ = ⟨n|ϱ| ˆ n⟩ =: P (n).
(5.23)
Employing (5.15) we can rewrite Eq. (5.12) in the following form
⟨f (φ)⟩ = Tr f (φ)ϱˆ
(5.24)
where f (φ) :=
π
f (φ)|φ⟩⟨φ|dφ.
(5.25)
−π
Therefore, to any classical function dependent on the phase, f (φ) one can assign the operator (5.25). For example if f (φ) = φ then the respective operator
φˆ =
π
φ|φ⟩⟨φ|dφ
(5.26)
−π
is the Garrison–Wong phase operator [43,58]. However, the crucial point is that [58]
k φˆ ̸= φk for k > 1.
(5.27)
φ and sin φ ̸= cos φˆ , So, for example, cos φ are the Susskind–Glogower operators [42,58] and cos ˆ sin φ ̸= sin φ . This can be easily understood from the fact that the mapping (5.4) which in terms of operators |φ⟩⟨φ|, φ ∈ [−π , π ) reads
M0 : B [−π , π ) ∋ X →
|φ⟩⟨φ|dφ
(5.28)
X
is a POV-measure but not a spectral measure [48,59,60]. However, the mapping E defined by (5.5) which in terms of |Θ ⟩ given by (4.3) reads E : B [−π , π ) ∋ X →
|Θ ⟩⟨Θ |dΘ
(5.29)
X
k
ˆ is a spectral measure. Consequently, in contrary to (5.27) one gets Θ
k for k = 0, 1, . . . . =Θ
Note that it is an easy matter to generalize the formula (5.25) on the case of any function f = f (φ, n), n = 0, 1, 2, . . . . First, observe that
ˆ Ω ˆ [Ks ](Θ , n) Π ˆ = 0ˆ , Π
for n = −1, −2, . . . .
(5.30)
Define
ˆ [Ks ](φ, n) := Π ˆ Ω ˆ [Ks ](−φ, n) Π ˆ, Ω
n = 0, 1, 2, . . . .
(5.31)
Given f = f (φ, n), n = 0, 1, 2, . . . we assign to it the following operator fˆ :=
∞ n =0
π
ˆ [Ks ](φ, n) dφ. f (φ, n)Ω
(5.32)
−π
One quickly finds that for f = f (φ) the formula (5.32) yields exactly (5.25) and for any f f (φ, n), n = 0, 1, 2, . . . we have ∞ ⟨fˆ ⟩ = Tr fˆ ϱˆ = n =0
π −π
f (φ, n)ϱ [Ks ](φ, n) dφ. W
=
(5.33)
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
Moreover, if f (φ, n) = fˆ =
∞
akl
k,l=0
kl
929
akl φ k nl then
1 k) . (φ k )ˆnl + nˆ l (φ 2
(5.34)
Therefore formula (5.32) can be considered as the generalized Weyl quantization prescription for number–phase functions leading to the symmetric ordering of operators. The same idea with another quantizer was proposed by J.A. Vaccaro [3]. We end this section with a remark which in a sense justifies our decision to name the quasiprobability function (5.11) or (5.21) the number–phase Wigner function for the state ϱˆ and the kernel KS . One quickly finds that under (5.30) and (5.31) 1 ˆ [KS ](φ, n)ϱˆ = Re ⟨φ|ϱ| Tr Ω ˆ n⟩⟨n|φ⟩ = ϱW [KS ](φ, n). 2π It shows that our prescription for obtaining the Wigner function follows the standard way. 6. Some examples of the number–phase Wigner functions 6.1. The Fock state |N ⟩ Inserting ϱˆ = |N ⟩⟨N | into (5.21) and employing (5.15) one gets
2 1 ϱW [KS ](φ, n) = ⟨φ|n⟩ δNn = δNn 2π
(6.1)
and this is just the Wigner function found by J.A. Vaccaro [3]. Then, the phase probability distribution (5.22) is now uniform P (φ) =
1
2π and the photon number probability distribution (5.23) is simply
P (n) = δNn
(6.2)
(6.3)
as should be for the Fock state |N ⟩. Note that ϱ [KS ](φ, n) = P (φ)P (n). W
6.2. Coherent states Consider the coherent state
1
2
|α⟩ = exp − |α| 2
∞ n=0
αn √ |n⟩, n!
C ∋ α = |α|eiϕ , ϕ ∈ [−π , π ).
(6.4)
The corresponding density matrix is
ϱˆ = |α⟩⟨α|.
(6.5)
Substituting (6.5) into (5.21) we obtain
|α|n exp − 21 |α|2 ϱW [KS ](φ, n) = Re ⟨φ|α⟩ exp[in(φ − ϕ)] √ 2π n! ∞ |α|2n exp −|α|2 |α|l = cos[l(φ − ϕ)] √ √ (l + n)! 2π n! l=−n ∞ |α|n exp −|α|2 |α|k = cos[n(φ − ϕ)] √ √ cos[k(φ − ϕ)] 2π n! k! k=0 ∞ |α|k + sin[n(φ − ϕ)] √ sin[k(φ − ϕ)] . k! k=0
(6.6)
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M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
Fig. 1. Plots of ϱ (0, n), n = 0, 1, 5, 10, 20, 40 as a function of α = |α|. W
a
b
c
Fig. 2. Exemplary plots of coherent state Wigner functions (a) α = 0.1, n = 0 and n = 1, (b) α = 1, n = 0, n = 1 and n = 2, (c) α = 5, n = 0, n = 1 and n = 2.
It is clear from (6.6) that if we put ϕ = 0 (as we do in all the plots) the coherent state Wigner functions assumes its maximum for φ = 0. In Fig. 1 we plot the dependence of this maximum on α = |α| for several values of n from 0 up to 40. We notice that as n grows the peak position shifts towards higher values of α . Moreover, slight change in a peak height is only observed for small values of n. Fig. 2 shows the coherent state number–phase Wigner functions for (a) α = 0.1, n = 0 and n = 1, (b) α = 1, n = 0, n = 1 and n = 3 and (c) α = 5, n = 0, n = 1 and n = 3. Fig. 3 shows coherent state Wigner functions for α = 5 and higher values of n, the sub-figure depicts the oscillatory character of the function. Finally, to show the increase of zeros with growing n, we plot the coherent state Wigner functions for α = 1 in Fig. 4 (a) n = 15, (b) n = 20 and (c) n = 25. 6.3. Squeezed states The amplitude of the Fock state |n⟩ for the squeezed state |α, ζ ⟩ is given as [57,61]
⟨n|α, ζ ⟩ = √
1
n!coshr
× Hn
1 iθ e tghr 2
n/2
α + α ∗ eiθ tghr 2eiθ tghr
exp −
1 2
∗ 2 iθ
|α| + (α ) e tghr 2
,
(6.7)
where Hn stands for the Hermite polynomial of degree n, C ∋ α = |α|eiϕ , ϕ ∈ [−π , π ) is the coherent amplitude and C ∋ ζ = reiθ , r ∈ R+ ∪ {0}, θ ∈ [−π , π ) is the squeeze parameter. Eq. (5.21) gives
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
931
Fig. 3. Coherent state Wigner functions for the fixed value of α (α = 5) and three different values of n (n = 15, n = 20 and n = 25). Note rapid oscillations around zero.
a
b
c
Fig. 4. Coherent state Wigner functions for α = 1, (a) n = 15, (b) n = 20 and (c) n = 25. Note the increase in the number of zeros with growing n.
following, rather complicated, expression for the Wigner function
ϱW [KS ](φ, n) =
1 2π coshr
× Hn
Re
1
√
n!
exp −|α|2 [1 + cos(2ϕ − θ )tghr ]
α ∗ + α e−iθ tghr
2e−iθ tghr
× Hk
α + α ∗ e−iθ tghr 2eiθ tghr
∞
1 ei(n−k)φ √ k! k=0
1 iθ e tghr 2
1 2
e−iθ tghr
n/2
k/2
.
(6.8)
In the case of squeezed vacuum (α = 0) formula (6.8) for 2n assumes the following form
√ (2n)! −tghr n ϱW [KS ](φ, 2n) = 2π coshr n! 2 √ ∞ (2l)! −tghr l × cos[(n − l)(2φ − θ )] l! 2 l =0 1
(6.9)
whereas for 2n + 1 it reads
ϱW [KS ](φ, 2n + 1) = 0.
(6.10)
Fig. 5(a) shows the squeezed vacuum number–phase Wigner function for n = 2 and the real squeeze parameter (ζ = r), r = 1, r = 0.8 and r = 0.6. Part (b) of Fig. 5 depicts the unsqueezed (θ = 0, r = 0) number–phase Wigner function for n = 0, n = 2 and n = 4.
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M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
a
b
Fig. 5. Squeezed vacuum Wigner functions (a) for n = 2 and three values of r (r = 1, r = 0.8 and r = 0.6), (b) for the fixed r = 1 and three values of n (n = 0, n = 2 and n = 4).
6.4. Black body radiation The density operator for a single mode of the black body radiation is given by
ϱˆ =
∞
1 − e−β }ω e−nβ }ω |n⟩⟨n|,
(6.11)
n =0
where β =
1 kT
and ω stands for the frequency of the mode. Inserting (6.11) into (5.21) one gets
1 1 − e−β }ω e−nβ }ω . 2π Then the phase and number probability distributions are ϱW [KS ](φ, n) = P (φ) =
1
(6.12)
(6.13)
2π
and −β }ω
P ( n) = 1 − e
−n β } ω
e
=
1
⟨n⟩ + 1
⟨n⟩ ⟨n⟩ + 1
n (6.14)
respectively, where ⟨n⟩ = exp(β1}ω)−1 . Observe that analogously as in the case of the Fock state one has ϱ [KS ](φ, n) = P (φ)P (n). W
6.5. The ‘Fock cat’ state We consider the ‘Fock cat’ state in the form [3]
|ψ FC ⟩ = cos η|N ⟩ + eiϕ sin η|N ′ ⟩.
(6.15)
Then
ϱˆ = |ψ FC ⟩⟨ψ FC | = cos2 η|N ⟩⟨N | + sin2 η|N ′ ⟩⟨N ′ | +
1
sin 2η eiϕ |N ′ ⟩⟨N | + e−iϕ |N ⟩⟨N ′ | .
2 Substituting (6.16) into (5.21) one gets
ϱW [Ks ](φ, n) =
1 2π
+
(6.16)
1 2
cos2 η δnN + sin2 η δnN ′
′ sin 2η cos (N − N )φ + ϕ (δnN + δnN ′ ) .
(6.17)
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
933
Fig. 6. The number–phase Wigner function for the ‘Fock cat’ state |ψ ⟩ = cos(π/10)|0⟩ + sin(π/10)|7⟩. FC
In Fig. 6 we plot the Wigner function for the ‘Fock cat’ state (6.15) with η = π /10, ϕ = 0, N = 0 and N ′ = 7. We conclude that in contrast to the case studied by J.A. Vaccaro [3], where an interference ring ′ is observed for some n between n = N and n = N′ , in our case the term with n = N interferes with 1 ′ N and vice versa by the term 2 sin 2η cos (N − N )φ + ϕ (δnN + δnN ′ ). Note that in case of η = 0 we reconstruct the Fock state |N ⟩ with ϱ [Ks ](φ, n) = 21π δnN . W
7. Concluding remarks The essential novelty of our work consists in obtaining the number–phase Wigner function from the Wigner function of a particle on the circle in a natural way by picking out the kernel K in an appropriate way. We have shown that using the concept of the enlarged Hilbert space for the Fock space HF one can easily define the number–phase Wigner function in quantum optics. In our case the enlarged Hilbert space of HF is the Hilbert space L2 (S 1 ). The advantage of such a choice is that it eliminates the need of using any unphysical phase space points with half integer values of the number of photons. Moreover, one can apply the elegant Weyl quantization formalism introduced for a particle on the circle S 1 . As a result we obtain a number–phase Wigner function which is real and its marginal distributions give the probability distributions of the phase and the number of photons, respectively. Further analysis and comparison of our results with the ones obtained by other authors will be given elsewhere. In particular, we intend to look for the number–phase Wigner function within the group-theoretical formalism developed by K.B. Wolf and co-workers (see e.g. [62–64]). This formalism enables us to define the Wigner function on any Lie group. In [63] the authors have found a Wigner function for the Euclidean group E (2). However, the relation between that Wigner function and our Wigner function on the cylinder given by Eq. (4.4) is rather not clear. We are going to deal with this question soon. Acknowledgement M.P. and J.T. were partially supported by the CONACYT (Mexico) grant no. 103478. References [1] [2] [3] [4] [5]
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From the Weyl quantization of a particle on the circle to number–phase Wigner functions Maciej Przanowski, Przemysław Brzykcy, Jaromir Tosiek ∗ Institute of Physics, Technical University of Łódź, Wólczańska 219, 90-924 Łódź, Poland
article
info
Article history: Received 14 April 2014 Accepted 18 October 2014 Available online 26 October 2014 keywords: Quantum phase Number–phase Wigner function
abstract A generalized Weyl quantization formalism for a particle on the circle is shown to supply an effective method for defining the number–phase Wigner function in quantum optics. A Wigner function for the state ϱˆ and the kernel K for a particle on the circle is defined and its properties are analysed. Then it is shown how this Wigner function can be easily modified to give the number–phase Wigner function in quantum optics. Some examples of such number–phase Wigner functions are considered. © 2014 Elsevier Inc. All rights reserved.
1. Introduction The present paper is a continuation of our previous work [1]. In fact, it has been motivated by a question raised by the referee of [1]. Namely, the referee pointed out that the Weyl quantization formalism developed in [1] should be closely related to the Wigner function and the Wigner representation of quantum phase investigated previously by the others [2–4]. So, here we follow this suggestion and we intend to display, how one can define the Wigner function which depends on the number and the phase. We arrive at this goal by employing the generalized Weyl quantization formalism for a particle on the circle given in [1]. As will be shown the answer to this question can be easily found and, moreover, it appears to be fairly natural within the generalized Weyl quantization machinery. It is well known that the Wigner function for a system of particles in R3 is a real function on the corresponding classical phase space so it depends on the Cartesian coordinates of the particles
∗
Corresponding author. E-mail addresses: [email protected] (M. Przanowski), [email protected] (P. Brzykcy), [email protected] (J. Tosiek). http://dx.doi.org/10.1016/j.aop.2014.10.011 0003-4916/© 2014 Elsevier Inc. All rights reserved.
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M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
and on the respective canonically conjugate momenta [5–7]. The Wigner function is uniquely defined by the density operator of the system and, conversely, the density operator is uniquely determined by the given Wigner function. Then, the expectation value of any quantum observable in a given quantum state can be found by integrating the product of the corresponding classical observable and the respective Wigner function. This procedure resembles very much the one well known in statistical physics. The only important difference consists in the fact that, in general, the Wigner function is not pointwise non-negative. Consequently, in general, it does not represent a probability distribution but, as is usually said, the Wigner function is a quasi-probability distribution. Nevertheless, the marginal distributions of any Wigner function give the probability distributions for coordinates and momenta, respectively. The natural question arises if one can define the analogous Wigner functions for the constraint quantum systems (e.g. a particle on the circle) or for the quantum systems described by the finite-dimensional Hilbert spaces. This question has raised a great deal of interest and many authors have analysed various examples of the Wigner functions for such quantum systems [8–19] (in particular see [18] and the references therein). Another interesting question, which turns out to be closely related to the previous one, concerns the definition of the Wigner function in quantum optics as a function of the photon number and the phase. As is known, the photon number and the phase can be considered as the canonically conjugated quantities and therefore, one attempts to find the Wigner function that depends on these quantities i.e. the so called number–phase Wigner function. This problem was explored by numerous authors. For example, J.A. Vaccaro and D.T. Pegg [20] defined a number–phase Wigner function employing the results of W.K. Wootters on the discrete Wigner functions in finite-dimensional spaces [13]. Such an approach is undoubtedly based on the celebrated Pegg–Barnett formalism in the theory of quantum phase [21–23]. A. Lukš with V. Peřinová [2], and M.R. Hush et al. [4] defined a number–phase Wigner function which was extended to rather unphysical half integer values of the number of photons. In the distinguished work by J.A. Vaccaro [3] a number–phase Wigner function is proposed assuming at the very beginning that this function should satisfy, besides the usual properties inflicted on Wigner function [7], some additional property leading to the interference fringes for the Wigner functions of the Schrödinger cat states. The aim of our paper is to show that the number–phase Wigner function can be easily defined with the use of the generalized Weyl quantization formalism on the cylindrical phase space S 1 × R1 under the observation that Hilbert space L2 (S 1 ) can be considered as an enlarged Fock space HF . The paper is organized as follows. In Section 2 we recall the main formulas of the generalized Weyl quantization formalism for the cylindrical phase space S 1 × R1 . The generalized Stratonovich–Weyl quantizer for an arbitrary kernel K is introduced and its properties are studied. The definition of the Wigner function (for an arbitrary kernel K ) in S 1 × R1 is given in Section 3. Main properties of the Wigner functions are also analysed. In Section 4 the Wigner function is specified to the case when the kernel K = KS leads to the symmetric ordering of operators under the Weyl quantization prescription. The number–phase Wigner function is defined in Section 5. We show there that having defined the Wigner function in S 1 × R1 for the kernel K = KS and employing the results of our previous work on quantum phase [1] one can quickly define a number–phase Wigner function in quantum optics. The properties of this Wigner functions are also studied in Section 5. Section 6 is devoted to some explicit examples of the number–phase Wigner functions. We consider the Fock states, the coherent states, the squeezed states, the black body radiation and the ‘Fock cat’ states. Finally, concluding remarks in Section 7 end the paper.
2. The generalized Weyl quantization on the cylinder Consider a particle on the circle S 1 . Let Θ ∈ [−π , π ) denote the angle coordinate of the particle (Important remark: In our paper the interval [−π , π ) is identified with the circle S 1 ) and L ∈ R1 be the angular momentum of this particle. The corresponding phase space is the cylinder S 1 × R1 . Given a function f = f (Θ , L) on S 1 × R1 one assigns to it an operator W [K ](f ) in the Hilbert space L2 (S 1 ) according to the rule [24,1] f = f (Θ , L) → W [K ](f )
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
W [K ](f ) :=
∞
π
K (σ , l)
−π
l=−∞
∞
n=−∞
π
921
f (Θ , n})
−π
dΘ dσ × exp −i(σ n + lΘ ) Uˆ (σ , l) 2π 2π ∞ π dΘ ˆ [K ](Θ , n) = f (Θ , n})Ω , 2π n=−∞ −π
(2.1)
where the function K = K (σ , l), σ ∈ [−π , π ), l ∈ Z is called the kernel and Uˆ (σ , l) is the unitary operator on the Hilbert space L2 (S 1 )
σ
Uˆ (σ , l) = exp i
ˆ Lˆ + lΘ
} i i ˆ exp σ l exp ilΘ σ Lˆ = exp 2 } i i ˆ = exp − σ l exp σ Lˆ exp ilΘ 2
=
∞
}
exp iσ
k+
k=−∞
l 2
|k + l⟩⟨k|,
σ ∈ [−π , π ).
(2.2)
The quantization rule (2.1) is called the generalized Weyl quantization with the kernel K . Moreover,
ˆ [K ](Θ , n) := Ω
π
∞ l=−∞
K (σ , l) exp −i σ n + lΘ
Uˆ (σ , l)
−π
dσ 2π
(2.3)
is the generalized Stratonovich–Weyl (GSW) quantizer for the kernel K . Recall that Uˆ (σ , l) has the following properties [17,25]
Tr Uˆ (σ , l) = 2π δl0 δ (S ) (σ ),
(2.4a)
Tr Uˆ Ď (σ , l)Uˆ (σ ′ , l′ ) = 2π δll′ δ (S ) (σ − σ ′ ),
(2.4b)
where δ (S ) (σ ) stands for the Dirac delta on the circle given by
δ (S ) (σ ) =
1
∞
2π l=−∞
exp ilσ .
(2.5)
As can be easily shown (see for example [6,1]), the natural assumptions about quantization impose some restrictions on the kernel K . Namely
ˆ ) for an arbitrary function f depending only on Θ , f = f (Θ ) iff (i) W [K ](f ) = f (Θ ∀l∈Z K (0, l) = 1,
(2.6a)
(ii) W [K ](f ) = f (Lˆ ) for an arbitrary function f depending only on L, f = f (L) iff
∀σ ∈[−π,π) K (σ , 0) = 1,
(2.6b)
(iii) the operator W [K ] is symmetric for any real function f = f (Θ , L) iff
∀σ ∈[−π,π), l∈Z K ∗ (σ , l) = K (−σ , −l) (where the star ‘∗’ stands for the complex conjugation).
(2.6c)
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Then the GSW quantizer (2.3) has the following important properties. (a) If (2.6a) or (2.6b) holds true then
ˆ [K ](Θ , n) = 1. Tr Ω
(2.7a)
(b) The condition (2.6c) yields
ˆ Ď [K (Θ , n)] = Ω ˆ [K (Θ , n)]. Ω
(2.7b)
(c)
ˆ [K ](Θ , n)Ω ˆ [K ](Θ ′ , n′ ) Tr Ω
=
1
∞
π
2π l=−∞ −π
K (σ , l)K (−σ , −l)
× exp i[σ (n − n′ ) + l(Θ − Θ ′ )] dσ .
(2.7c)
(d) If (2.6a) holds true then ∞
1
ˆ [K ](Θ , n) = |Θ ⟩⟨Θ |. Ω
(2.7d)
ˆ [K ](Θ , n)dΘ = |n⟩⟨n|. Ω
(2.7e)
2π n=−∞ (e) If (2.6b) holds true then
1 2π
π −π
Let fˆ be an operator in L2 (S 1 ) which arises from some function f (Θ , n}) on the quantized cylindrical phase space S 1 × Z by the generalized Weyl quantization prescription (2.1) and let gˆ be any operator in L2 (S 1 ). Then from (2.1) one quickly gets
Tr fˆ gˆ = Tr
∞
n=−∞
=
π
∞ n=−∞
π
ˆ [K ](Θ , n) f (Θ , n})Ω
−π
f (Θ , n})
−π
1 2π
dΘ
2π
gˆ
ˆ [K ](Θ , n)ˆg dΘ . Tr Ω
(2.8)
Two important remarks are to be made here. Remark 1. For K = 1 our generalized Weyl quantization rule (2.1) is equivalent to the one introduced by J.P. Bizarro in his distinguished paper [16] (see subsection C of the section III of [16]) and also to that considered by J.F. Plebański et al. in [17]. Then the generalized Weyl quantization rule (2.1) in a slightly different notation was given also by Rigas et al. [19] (see Eqs. (40) and (43) in [19]). However, one quickly realizes that the kernel K considered in [19] is necessarily of the form
K (σ , l) = exp i α(l, −σ ) −
1 2
σl ,
α∗ = α
(2.9)
which excludes the case of K (σ , l) = cos σ2l investigated in the following sections of the present work. Remark 2. The generalized Weyl quantization rule (2.1) with the kernel K satisfying (2.6a) for f (Θ , L) = Θ leads to the self-adjoint angle operator [26,1]
ˆ =i Θ
∞ (−1)j−k |j⟩⟨k|. j−k j,k=−∞
(2.10)
j̸=k
ˆ and sin Θ ˆ are self-adjoint operators which correspond to functions cos Θ Then the operators cos Θ ˆ , sin Θ ˆ and sin Θ , respectively. Moreover, the operators Lˆ , cos Θ
constitute the generators of the
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
923
Euclidean group E (2) and they play a crucial role in quantization on the cylinder proposed by K. Kowalski and J. Rembieliński [27] and H.A. Kastrup [28]. In our approach, which is founded on the generalized Weyl quantization prescription (2.1), the canonically conjugate variables are the angle ˆ respectively. We ˆ and L, Θ and the angular momentum L, which are represented by the operators Θ suppose that from the ‘point of view of a particle’ on the circle the angle Θ is more natural variable than cos Θ and sin Θ (or equivalently eiΘ ). 3. Wigner function Let ϱˆ be a density operator of a particle on the circle. It satisfies the usual conditions
ϱˆ + = ϱ, ˆ
(3.1a)
⟨ψ|ϱ|ψ⟩ ˆ ≥ 0 ∀|ψ⟩ ∈ L (S ), Tr ϱˆ = 1. 2
1
(3.1b) (3.1c)
For any observable represented by the operator fˆ the expectation value of this observable in the state ϱˆ is given by the well known formula
⟨fˆ ⟩ = Tr fˆ ϱˆ .
(3.2)
Assume that the operator fˆ arises from some classical observable f = f (Θ , L) as the result of the generalized Weyl quantization rule (2.1). So by (2.8) the relation (3.2) can be rewritten in the form
⟨fˆ ⟩ =
∞ n=−∞
π
f (Θ , n})
−π
1 2π
ˆ [K ](Θ , n)ϱˆ dΘ . Tr Ω
(3.3)
Introducing the function (see also [19])
ϱW [K ](Θ , n}) :=
1 2π
ˆ [K ](Θ , n)ϱˆ Tr Ω
(3.4)
which will be called the Wigner function for the state ϱˆ and the kernel K one writes (3.3) as
⟨fˆ ⟩ =
∞ n=−∞
π
f (Θ , n})ϱW [K ](Θ , n})dΘ .
(3.5)
−π
Observe that for K = 1 one gets the Wigner function introduced by M. Berry [8] and N. Mukunda [9,10] and then studied in all details by J.P. Bizarro [16]. The same function appears in [17,19,29]. Eq. (3.5) resembles very closely the fundamental formula from classical statistical mechanics defining the expectation value of the observable f = f (Θ , n}). Therefore we can identify ⟨fˆ ⟩ ≡ ⟨f (Θ , n})⟩. One easily finds that if the condition (2.6c) is fulfilled then by (2.7b) and (3.1a) we have ∗ ϱW [K ](Θ , n}) = ϱW [K ](Θ , n})
(3.6)
i.e. ϱW [K ] is a real function. Assume that (2.6a) holds true. Performing summation over n of both sides of (3.4) and employing (2.7d) one gets ∞
ϱW [K ](Θ , n}) = Tr |Θ ⟩⟨Θ |ϱˆ = ⟨Θ |ϱ| ˆ Θ ⟩ =: P (Θ ).
(3.7)
n=−∞
The function P (Θ ) given by (3.7) is the probability distribution of the angle Θ in the state ϱˆ . Analogously, assuming (2.6b), performing integration with respect to Θ of both sides of (3.4) and, finally employing (2.7e) one has
π
−π
ϱW [K ](Θ , n})dΘ = Tr |n⟩⟨n|ϱˆ = ⟨n|ϱ| ˆ n⟩ =: P (n})
(3.8)
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i.e. the probability distribution of the angular momentum L in the state ϱˆ . Note that any Wigner function should have the properties described by (3.6)–(3.8) [3,7,16]. It is worthwhile noticing that the discrete Wigner functions have been also extensively studied from the general point of view [29–34]. Moreover, a class of discrete-time Wigner functions has been introduced in signal processing [35], where Wigner distributions have been used for time–frequency analysis [36]. 4. Symmetric ordering of operators Wigner functions of a particle on the circle for K = 1 (the Weyl ordering) have been considered in [8–10,12,16,17,19,29,32,34]. In particular, recently [19] the Wigner functions corresponding to the coherent states on the circle have been studied. It should be noted that in [19] the definition of Wigner function (3.4) with the kernel K of the form (2.9) was given (in different notation) but further analysis has been carried out for K = 1. However, as shown in the next section, if one intends to carry over the results on the Wigner functions in the cylindrical phase space S 1 × R1 to the case of number–phase Wigner functions in quantum optics, it is more convenient to deal with the kernel which used in (2.1) leads to the symmetric ordering of operators. As it has been shown in [24,37–39,1] such a kernel reads
K (σ , l) = cos
σl
2
≡ KS (σ , l)
(4.1)
and it fulfils all the conditions (2.6a)–(2.6c). Therefore the relations (2.7a)–(2.7e) are also satisfied. Moreover, the respective Wigner function ϱW [KS ] has the basic properties (3.6)–(3.8). Substituting (4.1) into (2.3) one quickly gets
ˆ [KS ](Θ , n) = Ω
∞ 1
2 k=−∞
exp [−i(n − k)Θ ] |n⟩⟨k| + exp [i(n − k)Θ ] |k⟩⟨n|
= π |n⟩⟨n|Θ ⟩⟨Θ | + |Θ ⟩⟨Θ |n⟩⟨n| ,
(4.2)
where ∞
1
|Θ ⟩ = √
2π k=−∞
exp (−ikΘ ) |k⟩
(4.3)
ˆ and ⟨n|Θ ⟩ = √1 exp (−inΘ ) (for detailed is the normalized eigenvector of the angle operator Θ 2π
ˆ see for instance [26,1]). Inserting (4.2) into (3.4) we have analysis of the angle operator Θ ϱW [KS ](Θ , n}) =
1 2π
∞
Re
exp [−i(n − k)Θ ] ⟨k|ϱ| ˆ n⟩
k=−∞
= Re ⟨Θ |ϱ| ˆ n⟩⟨n|Θ ⟩ .
(4.4)
The function ϱW [KS ](Θ , n}) given by (4.4) defines the density operator ϱˆ uniquely. Indeed, one can rewrite (4.4) in the following form
ϱW [KS ](Θ , n}) =
1
∞
2π k=−∞
cos[(n − k)Θ ]Re⟨k|ϱ| ˆ n⟩ + sin[(n − k)Θ ]Im⟨k|ϱ| ˆ n⟩ .
(4.5)
From (4.5) we easily get Re⟨k|ϱ| ˆ n⟩ = 2
π
ϱW [KS ](Θ , n}) cos[(n − k)Θ ]dΘ for k ̸= n,
(4.6a)
−π
⟨n|ϱ| ˆ n⟩ =
π
ϱW [KS ](Θ , n})dΘ ,
(4.6b)
−π
Im⟨k|ϱ| ˆ n⟩ = 2
π −π
ϱW [KS ](Θ , n}) sin[(n − k)Θ ]dΘ
(4.6c)
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
925
√ (of course (4.6b) follows also from (3.8)). Multiplying both sides of (4.6c) by i = −1 and adding to (4.6a) one finally has
⟨k|ϱ| ˆ n⟩ = 2 ⟨n|ϱ| ˆ n⟩ =
π
ϱW [KS ](Θ , n}) exp[i(n − k)Θ ]dΘ for k ̸= n,
−π π
ϱW [KS ](Θ , n})dΘ .
(4.7)
−π
Concluding, the formulas (4.7) give all matrix elements of ϱˆ in the angular momentum basis {|n⟩}∞ ˆ itself. n=−∞ , ipso facto the density operator ϱ 5. Number–phase Wigner function The problem of defining the quantum phase of a harmonic oscillator or of a single-mode electromagnetic field has a long and involved history initiated by P.A.M. Dirac [40] and F. London [41] in the years 1926–27. We are not going to discuss all meanders of that history and we refer the reader to some of numerous works devoted to this question [21–23,42–58,1]. In the present paper we employ the results of our recent work [1], where it is argued that a reasonable way to define the quantum phase consists in extending the Fock space of the harmonic oscillator or the single-mode electromagnetic field to the Hilbert space L2 (S 1 ). This idea has been considered by several authors [49–53,56,58]. The construction given in Ref. [1] can ∞be stated as follows. Let HF be the respective Fock space and denote the Fock basis of HF by |n⟩ n=0 . (Remark: The vectors of HF will be marked
by the additional under-bar i.e. |n⟩, |ψ⟩, |χ⟩, . . . etc.) We embed HF in the Hilbert space L2 (S 1 ) by Jˆ : HF ∋
∞
cn |n⟩ →
n =0
∞
cn |n⟩ ∈ L2 (S 1 ),
cn ∈ C.
(5.1)
n = 0, 1, 2, . . . n = −1, −2, . . .
(5.2)
n =0
ˆ of L2 (S 1 ) onto HF by Define the projection Π 2 1 ˆ : L2 (S 1 ) ∋ |n⟩ → |n⟩ ∈ HF , Π L (S ) ∋ |n⟩ → |θ⟩ ∈ HF ,
where |θ ⟩ is the null vector in the space HF and put
Ď ˆ := Π ˆ |ψ⟩ ⟨ψ|Π Ď ⟨χ|Jˆ := Jˆ|χ⟩ .
(5.3a) (5.3b)
Then to any classical observable being a function of the phase φ , f = f (φ) one assigns the corresponding quantum observable in a state |ψ⟩ ∈ HF by quantizing the classical observable f (−Θ ) on the circle in the state Jˆ|ψ⟩ ∈ L2 (S 1 ) using the generalized Weyl quantization rule (2.1). The expectation value of the quantum observable corresponding to f = f (φ) in the state |ψ⟩ is equal to the expectation value of f (−Θ ) calculated for Jˆ|ψ⟩ ∈ L2 (S 1 ). [The substitution f (φ) → f (−Θ ) is justified by the commonly
ˆ (|Θ ⟩) = |−Θ ⟩.] As has been used conventions given by (4.3) and (5.15) which lead to the result Π shown in [1], the procedure described above is equivalent to the approach developed by J.H. Shapiro and S.R. Shepard [46], and by P. Bush, M. Grabowski and P.J. Lahti [48], where the quantum phase is given as the positive operator valued (POV) measure on [−π , π ) M0 : B [−π , π ) ∋ X →
∞ 1
2π j,k=0
exp [i(j − k)φ ] dφ X
|j⟩⟨k|
(5.4)
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M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
with B [−π , π ) standing for the family of Borel sets on [−π , π ). This POV measure is a compression ˆ E (X )Π ˆ of the spectral measure E i.e. M0 (X ) = Π E : B [−π , π ) ∋ X →
1
∞
2π j,k=−∞
exp [i(j − k)φ ] dφ
|j⟩⟨k|.
(5.5)
X
ˆ defined by (5.2) is just a Naimark projection [48,59,60]. Moreover, one can also Thus the projection Π show [1] that our procedure is equivalent to the Pegg–Barnett approach [21–23] but without introducing any, rather artificial, finite-dimensional Hilbert spaces. Now we are at the position, when the number–phase Wigner function can be defined. Let ϱˆ =
∞
ϱjk |j⟩⟨k|
(5.6)
j,k=0
be a density operator on the Fock space HF . Then the operator
ϱˆ = Jˆϱˆ Jˆ =
∞
ϱjk |j⟩⟨k|
(5.7)
j,k=0
is the density operator on L2 (S 1 ) associated to ϱˆ . Our idea is to define the number–phase Wigner function corresponding to the state ϱˆ (5.6) on HF by the Wigner function ϱW [K ](Θ , n}) corresponding to the state ϱˆ (5.7) on L2 (S 1 ). More precisely, the number–phase Wigner function for the state ϱˆ and the kernel K is defined as
ϱW [K ](φ, n) := ϱW [K ](−φ, n}) =
1 2π
ˆ [K ](−φ, n)ϱˆ , Tr Ω
n ≥ 0.
(5.8)
This approach based on Eq. (3.3) leads directly from the Stratonovich–Weyl quantizer to the Wigner function. It differs from the approach which starts from some axioms imposed on the Wigner function [3,7,16]. Since in HF , n = 0, 1, 2, . . . , for the sake of further consistency one must assume that the kernel K is such that
ˆ [K ](Θ , n)|k⟩ = 0. ∀Θ ∈[−π,π) , ∀j,k≥0, n<0 ⟨j|Ω
(5.9)
From (2.3) with (2.2) we easily infer that the condition (5.9) is equivalent to the following fundamental condition imposed on K
π
∀j,k≥0, n<0
K (σ , j − k) exp iσ
−π
j+k 2
−n
dσ = 0.
(5.10)
One quickly finds that this condition is not fulfilled for K = 1 (the Weyl ordering) and it is satisfied, for example, for K = KS . Therefore, we restrict ourselves to this latter case. From (5.8) applying (4.4), (5.6) and (5.7) one obtains
ϱW [KS ](φ, n) =
1 2π
Re
∞
ˆ n⟩ exp [i(n − k)φ ] ⟨k|ϱ|
k=0
= Re ⟨Θ = −φ|ϱ| ˆ n⟩⟨n|Θ = −φ⟩ ,
n = 0, 1, 2, . . . .
(5.11)
Let f = f (φ) be a classical observable relevant to the phase. Then employing (3.5), (5.8) and (5.11) we find that the expectation value of the quantum observable corresponding to f (φ) in the state ϱˆ given by (5.6) reads
⟨f (φ)⟩ =
∞ π 1
2π n,k=0 −π
f (φ) exp[i(n − k)φ]ϱkn dφ.
(5.12)
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
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In particular, if ϱˆ is a pure state
ϱˆ = |ψ⟩⟨ψ|,
⟨ψ|ψ⟩ = 1,
(5.13)
the formula (5.12) can be rewritten in the form ∞ π 1
⟨f (φ)⟩ =
2π n,k=0 −π
f (φ) exp[i(n − k)φ]⟨k|ψ⟩⟨ψ|n⟩dφ.
(5.14)
This result is in agreement with the respective result calculated within the Pegg–Barnett formalism (see Eq. (71) in [1]; note that in that equation in [1] the factor 21π is erroneously missing). Define the vector |φ⟩ in the rigged Hilbert space of HF by ∞
1
|φ⟩ := √
2π n=0
exp(inφ)|n⟩.
(5.15)
One quickly finds the relations
π
|φ⟩dφ⟨φ| = 1ˆ
(5.16)
−π
and
⟨φ|φ ′ ⟩ =
1 (S ) 1 i φ − φ′ δ (φ − φ ′ ) + − cot 2 4π 4π 2
(5.17)
(see [44]). From (5.16) we infer that any |ψ⟩ ∈ HF can be written in the form π
|ψ⟩ =
⟨φ|ψ⟩|φ⟩dφ.
(5.18)
−π
Then Eq. (5.14) can be rewritten as
⟨f (φ)⟩ =
π
−π
2
f (φ) ⟨φ|ψ⟩ dφ.
(5.19)
Consequently, the function ψ(φ) := ⟨φ|ψ⟩ can be considered as the wave function in the phase representation for the state |ψ⟩ and |⟨φ|ψ⟩|2 = |ψ(φ)|2 is the respective phase probability distribution. Thus, the vector |φ⟩ defined by (5.15) is the phase state vector as it is assumed in quantum optics (see e.g. [41,44,54–58]). Finally, in terms of |φ⟩ the formula (5.12) reads
⟨f (φ)⟩ =
π
ˆ dφ f (φ)⟨φ|ϱ|φ⟩
(5.20)
−π
and (5.11) takes the form analogous to (4.4)
ϱW [KS ](φ, n) = Re ⟨φ|ϱ| ˆ n⟩⟨n|φ⟩ ,
n = 0, 1, 2, . . . .
(5.21)
The marginal distributions calculated for the number–phase Wigner function (5.21) give the phase probability distribution ∞
ϱW [KS ](φ, n) = ⟨φ|ϱ|φ⟩ ˆ
n=0
=
1 2π
1 + 2Re
∞ k,n=0
exp[i(n − k)φ]ϱkn
=: P (φ)
(5.22)
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(compare with Eq. (6.86) from [57]) and the photon number probability distribution
π −π
ϱW [KS ](φ, n)dφ = ⟨n|ϱ| ˆ n⟩ =: P (n).
(5.23)
Employing (5.15) we can rewrite Eq. (5.12) in the following form
⟨f (φ)⟩ = Tr f (φ)ϱˆ
(5.24)
where f (φ) :=
π
f (φ)|φ⟩⟨φ|dφ.
(5.25)
−π
Therefore, to any classical function dependent on the phase, f (φ) one can assign the operator (5.25). For example if f (φ) = φ then the respective operator
φˆ =
π
φ|φ⟩⟨φ|dφ
(5.26)
−π
is the Garrison–Wong phase operator [43,58]. However, the crucial point is that [58]
k φˆ ̸= φk for k > 1.
(5.27)
φ and sin φ ̸= cos φˆ , So, for example, cos φ are the Susskind–Glogower operators [42,58] and cos ˆ sin φ ̸= sin φ . This can be easily understood from the fact that the mapping (5.4) which in terms of operators |φ⟩⟨φ|, φ ∈ [−π , π ) reads
M0 : B [−π , π ) ∋ X →
|φ⟩⟨φ|dφ
(5.28)
X
is a POV-measure but not a spectral measure [48,59,60]. However, the mapping E defined by (5.5) which in terms of |Θ ⟩ given by (4.3) reads E : B [−π , π ) ∋ X →
|Θ ⟩⟨Θ |dΘ
(5.29)
X
k
ˆ is a spectral measure. Consequently, in contrary to (5.27) one gets Θ
k for k = 0, 1, . . . . =Θ
Note that it is an easy matter to generalize the formula (5.25) on the case of any function f = f (φ, n), n = 0, 1, 2, . . . . First, observe that
ˆ Ω ˆ [Ks ](Θ , n) Π ˆ = 0ˆ , Π
for n = −1, −2, . . . .
(5.30)
Define
ˆ [Ks ](φ, n) := Π ˆ Ω ˆ [Ks ](−φ, n) Π ˆ, Ω
n = 0, 1, 2, . . . .
(5.31)
Given f = f (φ, n), n = 0, 1, 2, . . . we assign to it the following operator fˆ :=
∞ n =0
π
ˆ [Ks ](φ, n) dφ. f (φ, n)Ω
(5.32)
−π
One quickly finds that for f = f (φ) the formula (5.32) yields exactly (5.25) and for any f f (φ, n), n = 0, 1, 2, . . . we have ∞ ⟨fˆ ⟩ = Tr fˆ ϱˆ = n =0
π −π
f (φ, n)ϱ [Ks ](φ, n) dφ. W
=
(5.33)
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
Moreover, if f (φ, n) = fˆ =
∞
akl
k,l=0
kl
929
akl φ k nl then
1 k) . (φ k )ˆnl + nˆ l (φ 2
(5.34)
Therefore formula (5.32) can be considered as the generalized Weyl quantization prescription for number–phase functions leading to the symmetric ordering of operators. The same idea with another quantizer was proposed by J.A. Vaccaro [3]. We end this section with a remark which in a sense justifies our decision to name the quasiprobability function (5.11) or (5.21) the number–phase Wigner function for the state ϱˆ and the kernel KS . One quickly finds that under (5.30) and (5.31) 1 ˆ [KS ](φ, n)ϱˆ = Re ⟨φ|ϱ| Tr Ω ˆ n⟩⟨n|φ⟩ = ϱW [KS ](φ, n). 2π It shows that our prescription for obtaining the Wigner function follows the standard way. 6. Some examples of the number–phase Wigner functions 6.1. The Fock state |N ⟩ Inserting ϱˆ = |N ⟩⟨N | into (5.21) and employing (5.15) one gets
2 1 ϱW [KS ](φ, n) = ⟨φ|n⟩ δNn = δNn 2π
(6.1)
and this is just the Wigner function found by J.A. Vaccaro [3]. Then, the phase probability distribution (5.22) is now uniform P (φ) =
1
2π and the photon number probability distribution (5.23) is simply
P (n) = δNn
(6.2)
(6.3)
as should be for the Fock state |N ⟩. Note that ϱ [KS ](φ, n) = P (φ)P (n). W
6.2. Coherent states Consider the coherent state
1
2
|α⟩ = exp − |α| 2
∞ n=0
αn √ |n⟩, n!
C ∋ α = |α|eiϕ , ϕ ∈ [−π , π ).
(6.4)
The corresponding density matrix is
ϱˆ = |α⟩⟨α|.
(6.5)
Substituting (6.5) into (5.21) we obtain
|α|n exp − 21 |α|2 ϱW [KS ](φ, n) = Re ⟨φ|α⟩ exp[in(φ − ϕ)] √ 2π n! ∞ |α|2n exp −|α|2 |α|l = cos[l(φ − ϕ)] √ √ (l + n)! 2π n! l=−n ∞ |α|n exp −|α|2 |α|k = cos[n(φ − ϕ)] √ √ cos[k(φ − ϕ)] 2π n! k! k=0 ∞ |α|k + sin[n(φ − ϕ)] √ sin[k(φ − ϕ)] . k! k=0
(6.6)
930
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
Fig. 1. Plots of ϱ (0, n), n = 0, 1, 5, 10, 20, 40 as a function of α = |α|. W
a
b
c
Fig. 2. Exemplary plots of coherent state Wigner functions (a) α = 0.1, n = 0 and n = 1, (b) α = 1, n = 0, n = 1 and n = 2, (c) α = 5, n = 0, n = 1 and n = 2.
It is clear from (6.6) that if we put ϕ = 0 (as we do in all the plots) the coherent state Wigner functions assumes its maximum for φ = 0. In Fig. 1 we plot the dependence of this maximum on α = |α| for several values of n from 0 up to 40. We notice that as n grows the peak position shifts towards higher values of α . Moreover, slight change in a peak height is only observed for small values of n. Fig. 2 shows the coherent state number–phase Wigner functions for (a) α = 0.1, n = 0 and n = 1, (b) α = 1, n = 0, n = 1 and n = 3 and (c) α = 5, n = 0, n = 1 and n = 3. Fig. 3 shows coherent state Wigner functions for α = 5 and higher values of n, the sub-figure depicts the oscillatory character of the function. Finally, to show the increase of zeros with growing n, we plot the coherent state Wigner functions for α = 1 in Fig. 4 (a) n = 15, (b) n = 20 and (c) n = 25. 6.3. Squeezed states The amplitude of the Fock state |n⟩ for the squeezed state |α, ζ ⟩ is given as [57,61]
⟨n|α, ζ ⟩ = √
1
n!coshr
× Hn
1 iθ e tghr 2
n/2
α + α ∗ eiθ tghr 2eiθ tghr
exp −
1 2
∗ 2 iθ
|α| + (α ) e tghr 2
,
(6.7)
where Hn stands for the Hermite polynomial of degree n, C ∋ α = |α|eiϕ , ϕ ∈ [−π , π ) is the coherent amplitude and C ∋ ζ = reiθ , r ∈ R+ ∪ {0}, θ ∈ [−π , π ) is the squeeze parameter. Eq. (5.21) gives
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
931
Fig. 3. Coherent state Wigner functions for the fixed value of α (α = 5) and three different values of n (n = 15, n = 20 and n = 25). Note rapid oscillations around zero.
a
b
c
Fig. 4. Coherent state Wigner functions for α = 1, (a) n = 15, (b) n = 20 and (c) n = 25. Note the increase in the number of zeros with growing n.
following, rather complicated, expression for the Wigner function
ϱW [KS ](φ, n) =
1 2π coshr
× Hn
Re
1
√
n!
exp −|α|2 [1 + cos(2ϕ − θ )tghr ]
α ∗ + α e−iθ tghr
2e−iθ tghr
× Hk
α + α ∗ e−iθ tghr 2eiθ tghr
∞
1 ei(n−k)φ √ k! k=0
1 iθ e tghr 2
1 2
e−iθ tghr
n/2
k/2
.
(6.8)
In the case of squeezed vacuum (α = 0) formula (6.8) for 2n assumes the following form
√ (2n)! −tghr n ϱW [KS ](φ, 2n) = 2π coshr n! 2 √ ∞ (2l)! −tghr l × cos[(n − l)(2φ − θ )] l! 2 l =0 1
(6.9)
whereas for 2n + 1 it reads
ϱW [KS ](φ, 2n + 1) = 0.
(6.10)
Fig. 5(a) shows the squeezed vacuum number–phase Wigner function for n = 2 and the real squeeze parameter (ζ = r), r = 1, r = 0.8 and r = 0.6. Part (b) of Fig. 5 depicts the unsqueezed (θ = 0, r = 0) number–phase Wigner function for n = 0, n = 2 and n = 4.
932
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
a
b
Fig. 5. Squeezed vacuum Wigner functions (a) for n = 2 and three values of r (r = 1, r = 0.8 and r = 0.6), (b) for the fixed r = 1 and three values of n (n = 0, n = 2 and n = 4).
6.4. Black body radiation The density operator for a single mode of the black body radiation is given by
ϱˆ =
∞
1 − e−β }ω e−nβ }ω |n⟩⟨n|,
(6.11)
n =0
where β =
1 kT
and ω stands for the frequency of the mode. Inserting (6.11) into (5.21) one gets
1 1 − e−β }ω e−nβ }ω . 2π Then the phase and number probability distributions are ϱW [KS ](φ, n) = P (φ) =
1
(6.12)
(6.13)
2π
and −β }ω
P ( n) = 1 − e
−n β } ω
e
=
1
⟨n⟩ + 1
⟨n⟩ ⟨n⟩ + 1
n (6.14)
respectively, where ⟨n⟩ = exp(β1}ω)−1 . Observe that analogously as in the case of the Fock state one has ϱ [KS ](φ, n) = P (φ)P (n). W
6.5. The ‘Fock cat’ state We consider the ‘Fock cat’ state in the form [3]
|ψ FC ⟩ = cos η|N ⟩ + eiϕ sin η|N ′ ⟩.
(6.15)
Then
ϱˆ = |ψ FC ⟩⟨ψ FC | = cos2 η|N ⟩⟨N | + sin2 η|N ′ ⟩⟨N ′ | +
1
sin 2η eiϕ |N ′ ⟩⟨N | + e−iϕ |N ⟩⟨N ′ | .
2 Substituting (6.16) into (5.21) one gets
ϱW [Ks ](φ, n) =
1 2π
+
(6.16)
1 2
cos2 η δnN + sin2 η δnN ′
′ sin 2η cos (N − N )φ + ϕ (δnN + δnN ′ ) .
(6.17)
M. Przanowski et al. / Annals of Physics 351 (2014) 919–934
933
Fig. 6. The number–phase Wigner function for the ‘Fock cat’ state |ψ ⟩ = cos(π/10)|0⟩ + sin(π/10)|7⟩. FC
In Fig. 6 we plot the Wigner function for the ‘Fock cat’ state (6.15) with η = π /10, ϕ = 0, N = 0 and N ′ = 7. We conclude that in contrast to the case studied by J.A. Vaccaro [3], where an interference ring ′ is observed for some n between n = N and n = N′ , in our case the term with n = N interferes with 1 ′ N and vice versa by the term 2 sin 2η cos (N − N )φ + ϕ (δnN + δnN ′ ). Note that in case of η = 0 we reconstruct the Fock state |N ⟩ with ϱ [Ks ](φ, n) = 21π δnN . W
7. Concluding remarks The essential novelty of our work consists in obtaining the number–phase Wigner function from the Wigner function of a particle on the circle in a natural way by picking out the kernel K in an appropriate way. We have shown that using the concept of the enlarged Hilbert space for the Fock space HF one can easily define the number–phase Wigner function in quantum optics. In our case the enlarged Hilbert space of HF is the Hilbert space L2 (S 1 ). The advantage of such a choice is that it eliminates the need of using any unphysical phase space points with half integer values of the number of photons. Moreover, one can apply the elegant Weyl quantization formalism introduced for a particle on the circle S 1 . As a result we obtain a number–phase Wigner function which is real and its marginal distributions give the probability distributions of the phase and the number of photons, respectively. Further analysis and comparison of our results with the ones obtained by other authors will be given elsewhere. In particular, we intend to look for the number–phase Wigner function within the group-theoretical formalism developed by K.B. Wolf and co-workers (see e.g. [62–64]). This formalism enables us to define the Wigner function on any Lie group. In [63] the authors have found a Wigner function for the Euclidean group E (2). However, the relation between that Wigner function and our Wigner function on the cylinder given by Eq. (4.4) is rather not clear. We are going to deal with this question soon. Acknowledgement M.P. and J.T. were partially supported by the CONACYT (Mexico) grant no. 103478. References [1] [2] [3] [4] [5]
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