It is common knowledge that a quantum state at finite temperature is usually expressed by a mixed state ρ , ρ = e^{− β H}/Z, H is the system’ s Hamiltonian, and Z = tre^{− β H} is the partition function. In Ref.  [1] for conveniently calculating the ensemble average of an observable A in a mixed state Umezawa– Takahashi replaced the ensemble average

by an expectation-value of A in an appropriate pure state | ψ 〉

where | ψ 〉 is named a thermal vacuum state, and for different density operator ρ it corresponds to different | ψ 〉 . For example, for a chaotic field with^[2]

Umezawa– Takahashi found the corresponding thermal vacuum state (a pure state)^{[3, 4]}

where | 0, 0̃ 〉 is annihilated by ã and a, [ã , ã ^† ] = 1, and S(θ ) is the thermal operator, S(θ ) ≡ exp [θ (a^† ã ^† − aã )], which has the similar form to a two-mode squeezing operator except for the tilde mode in S(θ ) being a fictitious one, and θ is a parameter related to the temperature by tanh θ = exp (− ħ ω /2κ T), κ is the Boltzmann constant.

In our opinion, such a theory not only simplifies the statistical average calculation, but also provides an explicit two-mode state exhibiting quantum entanglement between system and reservoir. Then we are challenged by the question: What is the thermal vacuum state^{[5– 8]} for the squeezed chaotic light (SCL)? We shall employ the method of integration within ordered product (IWOP)^[9] of operators to find it, which can greatly simplify photon number average and quantum fluctuation calculation for SCL.

Our paper is arranged as follows. In Section 2, we briefly review the themal vacuum state theory and the partial trace method. In Section  3, we deduce the normal ordering form of the density operator for the squeezd chaotic light by using of the IWOP method, and obtain the thermal vacuum state corresponding to SCL, which turns out to be a one- and two- combinatorial mode squeezed state. As applications, in Sections  4– 5 we employ this new thermal vacuum state to calculate the photon number distribution and fluctuation, and the second-order degree of coherence. In Section  6 we derive the Wigner function of SCL.^{[10– 12]} A brief summary is given in the last section.

For a given real optical field described by density operator ρ , how can one get its thermal vacuum state | ψ (β )〉 ? In other words, which condition must be required for the thermal vacuum state | ψ (β )〉 such that its partial trace can lead to ρ ? Let the symbol Tr = tr represent performing trace both for real and ficitious modes, then we have

Since | ψ (β )〉 involves both real and fictitious modes, we should note that

Comparing Eq.  (5) with Eq.  (1) we see

which indicates that if we can find a pure state | ψ (β )〉 in the enlarged Hilbert space, then the partial trace over the tilde freedom of | ψ (β )〉 〈 ψ (β )| should lead to the density operator ρ of the system in a mixed state form.^[13]

For example, let us check if . By employing the completeness relation of coherent state in the fictitious space

and the normally ordered expansion of e^{λ a† a}

where : : denotes normal ordering, as well as the IWOP method we start from Eq.  (4) to derive

which reveals that performing a trace over the fictitious mode of | 0(β )〉 really leads to ρ _c of the chaotic optical field.

We now turn to the density operator of squeezed chaotic light^[14]

where

is the single-mode squeezing operator

and

In order to get the thermal vacuum state of ρ _s we first derive the normal ordering form of ρ _s with the help of the IWOP method. Using the Weyl expansion formula of an operator A in the Fock space^[15]

where denotes the Weyl ordering symbol,

is a coherent state, we have

According to the property that Weyl ordered operators are ordering-invariant under similar transformations, ^[15] thus using Eqs.  (16) and (13) we have

Thus replacing P, Q by p, q respectively, the classical Weyl correspondence of ρ _s is

which means that the classical correspondence of a Weyl ordered operator can be directly obtained by simply replacing Q ⟶ q, P ⟶ p. Then according to the Weyl correspondence rule, we have

where Δ (q, p) is the Wigner operator whose normal ordering form^[16] is

Substituting Eqs.  (18) and (20) into Eq.  (19), we can get the normal ordering form of density operator for the squeezed chaotic light

where

Using Eq.  (15) we can re-express Eq.  (21) as

Then we introduce the real mode coherent state

with the help of the IWOP method and using the integration formula

we can rewrite Eq.  (23) as

Noticing : e^{− a† a} : = | 0〉 〈 0| and Eq.  (24), we can rewrite Eq.  (26) as

Further, using 〈 0̃ | z̃ 〉 = e^{− | z| 2/2}, | z̃ 〉 is the coherent state^{[17, 18]} in the fictitious mode, equation  (27) can be put into the form

thus according to Eq.  (7) we conclude

this is the thermal vacuum state corresponding to SCL. From another point of view, | ψ (β )〉 _s in Eq.  (29) is a kind of one- and two-mode combinatorial squeezed state in real– fictitious mode Fock space.

In particular, when r = 0, there is no squeezing, from Eq.  (22) we see , then | ψ (β )〉 _s reduces to | 0(β )〉

which means that ρ _s reduces to ρ _c, as expected.

The pure state | ψ (β )〉 _s is normalized and can be shown as follows:

this is equivalent to

The introduction of | ψ (β )〉 _s can greatly simplify the calculation of the statistical average. For instance, we calculate photon-number distribution in SCL, using Eq.  (29) and the coherent state representation ∫ (d²z/π ) | z〉 〈 z| = 1 we derive

Thus the average photon number is

which indicates that the average photon number increases with the squeezing parameter r.

Also using | ψ (β )〉 _s we calculate

It follows that

then the photon number fluctuation is

this result seems to be newly reported, thus we see that for the chaotic state, the stronger the squeezing is, the larger the fluctuation is. This is a new property of chaotic light. We have derived it by virtue of the thermal vacuum state | ψ (β )〉 _s.

Using Eqs.  (35) and (36) we further examine the second-order coherence degree^{[19, 20]}

which indicates that the squeezed chaotic light is bunching, and when , it reduces to chaotic light, exhibiting the bunching effect, thus the squeezed chaotic light is still classic.

The Wigner function of | ψ (β )〉 _s is

where Δ (α ) is the Wigner operator, its coherent state representation is

Substituting Eq.  (41) into Eq.  (40), using the completeness of the coherent state and the integration formula

we obtain

Due to Eq.  (22), equation  (43) can be re-written as

In summary, by virtue of the partial trace method and the IWOP technique, we obtained the thermal vacuum state for the squeezed chaotic light. It helps us to successfully calculate the photon number disturibution, quantum fluctuation, and the second-order coherence degree, which shows that the squeezed chaotic light is still of a classic nature. For the Radon transformation property of squeezed chaotic light we can refer to Ref.  [21]. The thermal vacuum state corresponding to squeezed chaotic light we found in this work may bring convenience for studying quantum decoherence of a squeezed state in the amplitude damping channel and quantum controlling of squeezing.