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Hydrodynamic interpretation of generic squeezed coherent states:

A kinetic theory

Nezihe Uzun∗

Univ Lyon, Ens de Lyon, Univ Lyon1, CNRS,

Centre de Recherche Astrophysique de Lyon UMR5574, F69007, Lyon, France

(Dated: October 5, 2021)

The hydrodynamic interpretation of quantum mechanics treats a system of particles in an effec-tive manner. This interpretation is known to be mathematically equivalent to the de Broglie–Bohmtheory and together with the Wigner–Weyl–Moyal approach, it allows one to study a system ina statistical fashion. In this work, we investigate squeezed coherent states within the hydrody-namic interpretation. The Hamiltonian operator in question is time dependent, n–dimensional andin quadratic order. We start with deriving a phase space Wigner probability distribution and anassociated equilibrium entropy for the squeezed coherent states. Then, by following Moyal’s sta-tistical arguments, we decompose the joint phase space distribution into two portions: a marginalposition distribution and a momentum distribution that is conditioned on the post–selection of po-sitions. Our conditionally averaged momenta is shown to be equal to the Bohm’s momenta whoseconnection to the weak measurements is already known in the literature. In the mean time, wekeep track of the corresponding classical system evolution by identifying shear, magnification androtation components of the symplectic phase space dynamics. This allows us to pinpoint whichportion of the underlying classical motion appears in which quantum statistical concept. We showthat our probability distributions satisfy the Fokker–Planck equations exactly and they can be usedto decompose the equilibrium entropy into the missing information in positions and in momentaas in the Sackur–Tetrode entropy of the classical kinetic theory. Eventually, we define a quantumpressure, a quantum temperature and a quantum internal energy which are related to each other inthe same fashion as in the classical kinetic theory. We show that the quantum potential incorporatesthe kinetic part of the internal energy and the fluctuations around it. This allows us to suggest aquantum conditional virial relation. In the end, we show that the kinetic internal energy is linkedto the fractional Fourier transformer part of the underlying classical dynamics similar to the casewhere the energy of a quantum oscillator is linked to its Maslov index.

I. INTRODUCTION

The search for the underlying ties betweenclassical and quantum theories has been a longquest without a unique solution. Nevertheless,for a given quantum system, certain states canbe linked to its classical dynamics. For exam-ple, the wave function of a coherent state rep-resents a system with minimum uncertainty inwhich the expectation values of the position andmomentum operators follow classical trajecto-ries. This is the closest one can get to a clas-sical picture which was the original motivationof Schrodinger when he first discovered those

∗ nezihe.uzun@ens-lyon.fr

states [1]. The name “coherent” was given byGlauber when he extended Schrodinger’s workfor quantum optics in order to study the co-herency properties of light within the realmof standard harmonic oscillators [2, 3]. Sincethen, the definition of those states have beenextended for more generic systems [4–6]. Theyare specifically useful when studying the semi-classical limit of generic systems, or the oneswhich can be modelled by quadratic Hamil-tonians [5]. Applications of squeezed coher-ent states are mostly known to the researcherswithin the quantum optics and quantum in-formation processing fields. Those states areknown to reduce the quantum mechanical noiseand to increase the sensitivity in interferomet-ric measurements [7–10]. This makes them per-fect candidates for real life experiments that re-

2

quire highest level of precision. In addition, theextensive application area of squeezed coherentstates provides links to even wider perspectivesthrough their usage in gravitational wave detec-tion [11–13], in bio–imaging [14, 15] and even inthe early universe cosmology [16–18].

Alternatively, there exists a formalism innon–relativistic quantum mechanics that re-lates the wave function of a given systemto the Hamilton–Jacobi equations of classicaltrajectories. This is known as causal/pilotwave/deterministic/the de Broglie–Bohm the-ory. In the literature, it is usually attributedto the work of Bohm [19, 20], due to his majorcontribution and him reviving the pilot wavetheory of de Broglie [21, 22] quite outspokenly.The early works of Madelung [23], in additionto the contributions of Takabayasi [24] are usu-ally overlooked. Even though Madelung’s, deBroglie’s and Bohm’s approaches are mathe-matically equivalent, their ontologies seem to bedifferent. According to the de Broglie–Bohm in-terpretation, particles follow trajectories guidedby a pilot wave whose wave function has a phys-ical connotation. This is true even for a singleparticle. However, in the hydrodynamic inter-pretation of Madelung, the idea is to considermany particles in which the corresponding wavefunction of the system reflects an effective dy-namics. In both of those interpretations, nev-ertheless, it is the so-called quantum potential

that is responsible for the underlying quantumphenomena.

Another course of action is the Wigner–Weyl–Moyal approach which allows one to define aquasi–probability distribution on a phase spaceanalogous to the one in classical mechanics. Forpure states, for example, Wigner function [25]is shown to be the Weyl symbol of the densityoperator. This allows one to have a clear pas-sage to classical physics as the Wigner functionis used to obtain the expectation values of op-erators on phase space. Mainly, when combinedwith the Moyal product and the Moyal bracket[26], one recovers the von Neumann equationwhich reduces to the Liouville equation in theclassical limit [27]. This is at the core of the de-formation quantization which provides connec-

tions between classical and quantum physics.

On the other hand, finding the exact solu-tions for time dependent systems both in clas-sical and in quantum mechanics is not an easytask. As the invariants provide symmetries ofa given system, the Lewis–Riesenfeld invariantmethod [28, 29] has been used widely in the lit-erature in order to find the exact solutions oftime dependent systems. The classical corre-spondent of this quantum operator is known tobe an invariant of the corresponding classicalsystem. Unfortunately, the Lewis–Riesenfeldinvariant is usually considered as a mathemati-cal tool. Its physical and/or geometric interpre-tation are not studied in detail in the literature.

All of those approaches listed above providemeans to relate classical and quantum mechan-ics. Naturally, the links between some of thoseformalisms have been already established. Forexample, one of the most concise investigationsof the squeezed coherent states that makes useof the Wigner–Weyl–Moyal approach was givenby Littlejohn [5]. Moreover, the relationshipbetween the Wigner–Weyl–Moyal approach andthe de Broglie–Madelung–Bohm interpretationhas been investigated before [30–33]. The Gaus-sian states of a harmonic oscillator [34–36] anda particle in 1-dimensional Poschl–Teller po-tential [35] have also been investigated withinthe de Broglie–Madelung–Bohm theory. In ad-dition, the Lewis–Riesenfeld invariant of theGaussian states was established in [37] for onedimensional systems. However, there exists nostudy that unites all of the approaches men-tioned above in order to present a full picture.

In this work, we investigate time depen-dent squeezed coherent states in n–dimensionswithin a non–relativistic setting. It is knownthat a coherent state is an eigenstate of theannihilation operator. Thus, in general, thereis a common practice to study them by mak-ing use of the ladder operators and the numberstates. We will not follow this route as our aimis to study a system within the hydrodynamicinterpretation. We would like to keep trackof the classical and the quantum phase spacevariables that are directly meaningful for mea-surements. For this, the Wigner–Weyl–Moyal

3

formalism and the de Broglie–Madelung–Bohminterpretation will be predominant in our con-struction.

Ultimately, we want to show that oncethe statistical arguments are adopted properlywithin a hydrodynamic interpretation, a ki-netic theory and certain thermodynamic vari-ables can be defined exactly for a quadratic sys-tem in question. What is more profound is thatit is the underlying symplectic, classical dynam-ics that guides the quantum evolution of thesqueezed coherent states and the correspondingthermodynamic variables.

In order to achieve this, we start Section (II)by providing some preliminaries. We give a briefsummary of the de Broglie–Madelung–Bohmformalism. We then introduce a symplecticphase space for convenience as it is the startingpoint for the investigation of generic squeezedcoherent states. Next, we briefly summarizeLittlejohn’s construction [5] on the squeezed co-herent states. The preliminaries section endswith the introduction of the Wigner functionand its associated covariance matrix. Those willbe important for identifying statistical and ther-modynamical concepts.

In Section (III), we provide the main bodyof our own contribution on hydrodynamic in-terpretation of the generic squeezed coherentstates. For this, in Section (III A), we pro-vide the foundations of our construction. Westart with the polar decomposition of the Gaus-sian squeezed coherent state wave functionthat is exact. This allows us to decomposethe Schrodinger equation into pure real andimaginary parts which is the starting pointof the de Broglie–Madelung–Bohm approach.Then, we present the associated Wigner func-tion and identify its covariance matrix in Sec-tion (III A 2). In the mean time, we keep trackof the classical phase space shears, magnifica-tions and rotations in order to identify whichportion is responsible for which quantum sta-tistical phenomenon in phase space.

In Section (III B), motivated by Moyal’s sta-tistical approach [26], we identify three types ofprobabilities: (i) a joint distribution, i.e., theWigner function, which is defined on the en-

tire phase space; (ii) a marginal distributionon position space; (iii) a conditional momen-tum distribution shaped by the post–selectionof the positions. Note that all of the proba-bility distributions listed above take Gaussianforms as the wave function in question is alsoGaussian. We then make use of the Wignerdistribution in order to define a phase spaceShannon entropy. As the coherent states areknown to be minimum uncertainty states, ourphase space entropy takes an extremum valuethroughout the evolution of the system. Thisallows us to study equilibrium thermodynamicsin a dynamical sense within a hydrodynamic in-terpretation.

Before introducing the full thermodynamicanalysis, we derive the Fokker–Planck equationfor the probability densities in Section (III B 3).We show that it is not only the marginal po-sition distribution that satisfies a continuityequation but all probability distributions. Wediscuss the probability flux related to the ro-tational degrees of freedom in addition to theone of the linear flow. Note that those Fokker–Planck equations are applicable for a dynami-cal situation unlike the standard case which wasoriginally derived for stationary scenarios.

We return back to the thermodynamic anal-ysis in Section (III C), in which we start bypresenting the analogy between the quantumphase space entropy and the Sackur–Tetrodeentropy that was originally derived for the clas-sical kinetic theory. Then, we follow Sonego’sdefinitions in [38] in order to obtain a quan-tum pressure and a quantum temperature forthe squeezed coherent states. Next, we dis-cuss the internal energy and its kinetic partthat takes a similar form as in the classical ki-netic theory. We show its relation to the quan-tum potential that is the key element of the deBroglie–Madelung–Bohm approach. Namely,we demonstrate that the quantum potential rep-resents the kinetic internal energy of the systemand the fluctuations around it at equilibrium.We also suggest a quantum virial relation whichassociates the conditional kinetic energy to aquantum potential energy term sourced solelyby the quantum potential. In the end, we pro-

4

vide the link between the internal kinetic energyand the Maslov index defined for the symplec-tic paths. Essentially, we show that the quan-tum kinetic internal energy of a system is linkedto the fractional Fourier transformations of thecorresponding classical trajectories even if thesystem in question does not have periodic or-bits.

Finally, in Section (IV), we provide a sum-mary of our investigation, in addition to discus-sions regarding the extension of the domain ofapplicability of the current results.

II. PRELIMINARIES

A. The de Broglie–Madelung–Bohm

approach

There exists a correspondence between thepaths taken by quantum particles and the pathstaken by classical particles within the trajectoryapproach of the de Broglie–Madelung–Bohm[19–23]. Depending on the interpretation, thosetrajectories either reflect a physical, tractabletrajectory of a particle or an effective, meanstream–line trajectory of an ensemble of par-ticles. The cost that has to be paid in returnis the introduction of hidden variables to thetheory whose existence has been debated in theliterature many times.

Let us now introduce the summary of thecausal theory in its original version. For this,we will assume that there exists a particle withmass m to which a complex wave function isassigned in its polar form

ψ = R exp

(iS

~

), (1)

where ~ = h/(2π) with h being the Planck’sconstant, R = R (q, t) is a real amplitude andS = S (q, t) is a real phase function. In thecausal interpretation, the wave function satisfiesthe Schrodinger equation in the following form

i~∂ψ

∂t=

(− ~2

2m∇2 + V (q, t)

)ψ, (2)

where ∇ = ∂/∂q and V (q, t) is the classicalpotential. In general, V is a generic functionof positions and it has no momentum depen-dence. Substitution of the wave function in itspolar form, eq. (1), into the Schrodinger equa-tion given in the form in eq. (2), results in acomplex equation. Its pure imaginary and purereal parts are written respectively as

∂R

∂t= − 1

2m

[R∇2

S + 2∇R ·∇S], (3)

∂S

∂t= −

[(∇S )2

2m+ V (q, t) − ~2

2m

∇2R

R

].(4)

The equations above have been interpreted ina hyrodynamic realm due to two main reasons.Firstly, one can define a flux–like term, j, thatis associated with a wave function,

j =~

2mi(ψ∗

∇ψ − ψ∇ψ∗) = R2∇S

m, (5)

where a probability density is defined throughρ = [R (q, t)]

2and a velocity term is given by

v = ∇S /m. In that case j = ρv holds. Then,the imaginary part of the Schrodinger equation,eq. (3), can be viewed as a continuity equation,

∂ρ

∂t+ ∇. (ρv) = 0. (6)

Secondly, when ~ → 0, the real part of theSchrodinger equation, (4), gives the Hamilton–Jacobi equation of the classical mechanics, i.e.,

−∂S

∂t=

(∇S )2

2m+ V (q, t) = H, (7)

with S playing the role of the action functional.

In the quantum case, the term

Q (q, t) = − ~2

2m

∇2R

R(8)

is non–zero and it appears in the quantumHamilton–Jacobi equation in the same form asthe classical potential, V (q, t). That is whyit is known as the quantum potential which isresponsible for the quantum behaviour of thegiven particles. The equation of motion now

5

follows as

dp

dt= −∇ (V +Q) , (9)

where the momentum p = mv has differentconceptualizations depending on the chosen in-terpretation. For instance, it might correspondto a single particle momentum within the deBroglie–Bohm theory. Whereas, in Madelung’shydrodynamic interpretation [23] it correspondsto an effective momentum associated with an ir-rotational continuous “fluid” of particles.

Likewise, the quantum potential has dif-ferent interpretations as well. According toMadelung, for example, the quantum potentialis attributed to some quantum internal forces ofa fluid. Alternatively, within a thermodynamicinterpretation, the quantum potential can berelated to the averaged kinetic energy of thequantum particles, the temperature of the cor-responding system and the thermal vacuum en-ergy [39]. Similarly, it can be interpreted as theinternal energy of a system [40]. In certain in-vestigations, quantum potential acts as an agentthat allows the interchange of information be-tween systems [41, 42]. In the quantum cosmo-logical realm, Bohmian interpretation and thequantum potential can even be related it to thedark energy problem [43].

In summary, quantum potential allows oneto identify the physical phenomena behind thequantum behaviour of a system. In Sec-tion (III C 3), we will derive it for squeezed co-herent states of time dependent systems in n–dimensions and we will interpret it thermody-namically similar to the ones in [39, 40].

B. Symplectic phase space of classical

orbits and the quadratic Hamiltonians

Let us now set up the phase space of a clas-sical system by defining positions qa ∈ R

n andmomenta pa ∈ Rn with a, b = 1...n as inde-pendent variables. We will consider only thoseHamiltonians that are homogeneous quadraticfunctions of q’s and p’s which will be inter-preted as canonical phase space coordinates.

For a classical system, consider a 2n–dimensional phase space N(R2n, ω) that is en-dowed with a symplectic form ω and Darbouxcoordinates zi = (qa, pb)

⊺. Here, i, j =1...2n and ⊺ refers to the transpose operator.

We write the Poisson bracket of two functionsf and g as

f, g =∂f

∂ziωij ∂g

∂zj, (10)

where ω is called the fundamental symplectic

matrix. It is defined through

zi, zj = ωij , ωij =

[0n In−In 0n

],(11)

where In and 0n are n–dimensional identity andzero matrices, respectively.

The matrix ω satisfies

ω⊺ = ω−1 = −ω, ω2 = −I2n, detω = 1.

(12)

Here, the inverse operator is denoted by −1 andfor the determinant of a matrix, we use “det”.Given this, the symplectic two form acting ontwo arbitrary phase space vectors z and z′ canbe written as

ω (z, z′) = z⊺ω−1z′ = p⊺ q′ − q⊺p′. (13)

We will now choose the Hamiltonian function,H , to be time dependent, i.e., H = H (z, t) witht ∈ R and quadratic in z. Then, H is closedunder the Poisson bracket (10) and thus forma Lie algebra. Let us denote the Lie operatorcorresponding to H as

LH [] = −H,

(14)

which has a 2n× 2n Hamiltonian matrix repre-sentation that we will denote by

LH =

[b⊺(t) c(t)−a(t) −b(t)

], (15)

where a = a⊺, c = c⊺ and b are all n×n dimen-sional, time dependent, arbitrary matrices witha, c > 0n and ac − b2 > 0n. We will denote

6

the set of all 2n× 2n Hamiltonian matrices as

h := LH ∈ R2n×2n| (ωLH)

⊺= (ωLH).(16)

Now we write the Hamiltonian function as

H (z, t) =1

2z⊺ω⊺LHz, (17)

such that the Hamiltonian equations are

LH

[zi]

= −H, zi = ωij ∂H

∂zj, (18)

or simply,

dz

dt= LHz. (19)

Given the initial conditions z0 and t0 = 0, thesolution of eq. (19) is given by

z = S(t)z0, (20)

where S is obtained by taking the exponentialmap of the Hamiltonian matrix LH. Therefore,S is a 2n× 2n symplectic matrix satisfying

S⊺ ω S = ω, detS = 1. (21)

Note that due to eqs. (19) and (20), S also fol-lows the Hamiltonian flow, such that,

dS

dt= LHS, (22)

holds for the initial conditions S0 = I2n. Let uswrite this matrix in the block form

S(t) =

[A(t) B(t)C(t) D(t)

], (23)

where A, B, C and D are all n × n matricessatisfying

A⊺C, B⊺D, AB⊺, CD⊺ ⇒ symmetric,(24)

A⊺D−C⊺B = In, (25)

AD⊺ −BC⊺ = In, (26)

due to the symplecticity conditions (21). Thosematrices form the symplectic group Sp (2n,R)which has crucial importance for classicalquadratic systems and their quantization. Wewill denote the set of all real 2n×2n symplectic

matrices as

s := S ∈ R2n×2n|S⊺ω S = ω. (27)

In order to understand and identify the effectof the linear symplectic transformation on theevolution of phase space variables, one can usecertain techniques to decompose symplectic ma-trices into its submatrices. However, not all ofthose decompositions are unique. On the otherhand, Iwasawa showed that any symplectic ma-trix belonging to Sp(2,R) can be decomposeduniquely into its nilpotent subgroup, an abeliansubgroup and a maximally compact subgroup[44]. Those correspond to shearing, magnifica-tion and rotation effects on the phase space co-ordinates respectively. In the optics community,for example, those matrices represent lenses,magnifiers and fractional Fourier transformers.

Later, the Iwasawa decomposition is gener-alized to higher order symplectic matrices inwhich case the matrices responsible for the mag-nification effect do not form a group. Therefore,one refers to it as a factorization of the sym-plectic matrix or a modified–Iwasawa decompo-sition. It is given as [45, 46]

S =

[A B

C D

]

=

[I2 02

−g I2

] [s 02

02 s−1

] [Reu Imu

−Imu Reu

]

= l(g) m(s) f(u) (28)

where l(g) represents the shearing or lensingin phase space, m(s) represents magnificationsand f(u), being a fractional Fourier trans-former, represents rotation–like effects. Here,the n×n matrices that appear in (28) are givenin terms of the sub–blocks of the symplectic ma-trix, S, as

g = − (CA⊺ + DB⊺) (AA⊺ + BB⊺)−1

= g⊺,

s = (AA⊺ + BB⊺)1/2

= s⊺,

u = (AA⊺ + BB⊺)−1/2

(A + iB) ∈ U(n). (29)

In the following sections we will see that the

7

modified Iwasawa factorization is useful in iden-tifying how the different factors of the classi-cal phase space evolution find their correspon-dences in the quantum mechanical evolution.

In the next section, we give a brief summaryof the squeezed coherent states which is mostlybased on Littlejohn’s construction [5]. This sec-tion involves only those points that are imme-diately relevant for us which does not give thefull credit to the original paper.

C. Generic squeezed coherent states

Let us consider the quantum counterpartsof the classical positions and momenta, rep-resented in the position space as, q =(q1, q2, ..., qn)⊺ and p = (p1, p2, ..., pn)⊺, respec-tively.

They operate on a function f as

q [f ] = f · q, (30)

p [f ] = −i~∂ f∂q

, (31)

where the “ · ” denotes the standard multiplica-tion. They satisfy the Heisenberg commutationrule

[pk, qj ] = −i~δkj , (32)

where δkj is the Kronecker delta. Likewise,the quantum Hamiltonian operator which is thecounterpart of the quadratic classical Hamilto-nian given in eq. (17) is given by

H(t) =1

2z†ω⊺LH(t)z with z =

[q

p

],(33)

where “†” denotes the conjugate transpose andz is the quantum counterpart of the phasespace vector z. For the quadratic system givenin eq. (33), the expectation value of z fol-lows the underlying classical trajectory due tothe Ehrenfest Theorem. This means 〈z〉 =〈Ψ| z |Ψ〉 = z, where |Ψ〉 is the state which hasa wave function Ψ.

In order to define and study the evolutionof the wave functions of the squeezed coher-

ent states, one considers two sets of operations:(i) translations given by the Weyl–Heisenbergoperators and (ii) squeezings generated by themetaplectic operators.

The translation operator, T , is given withinthe Weyl–Heisenberg algebra. It is responsi-ble for translating a given object. For instance,translation of z by an amount z′ is given by

T † (z′ ) z T (z′) = z + z′, (34)

such that

〈Ψ| T † (z′) z T (z′) |Ψ〉 = z + z′. (35)

Formally, they are defined as

T (z′) = exp

(i

~ω [z, z′]

)

= exp

(− i

~(p′ · q− q′ · p)

), (36)

where the operator ω [z, z′] acts as a quantumcounterpart of the symplectic two–form ω (z, z′)

given in eq. (13). The operator T is unitary andit satisfies

T−1 (z′) = T (−z′) = T † (z′) . (37)

However, its product rule follows as

T (z) T (z′) = exp

(i

2~ω (z, z′)

)T (z + z′) ,

(38)

and thus those operators do not form a groupdue to the phase factor. Nevertheless, they canbe used to generate a propagator that evolvesthe quantum expectation values on the classicaltrajectory.

Imagine segmenting the classical trajectoryinto N straight pieces with a correspondingtranslation operator for each segment. By (i)making use of their products, i.e.,

T (zN − zN−1) ... T (z2 − z1) T (z1 − z0) ,

(39)

(ii) using the product rule in eq. (38), and (iii)considering the antisymmetry of the symplectic

8

two–form, one can obtain a propagator corre-sponding to T as [5],

UT (t) = exp

(− i

2~

∫ t

0

ω (z, z) dt

)T (z(t)) T † (z0) ,

(40)

in the limit N → ∞ and zN = z(t). Here,the overdot denotes the standard total deriva-tive with respect to time parameter. Note thatthe propagator UT (t) takes a Gaussian coherentstate to another Gaussian coherent state witha different expectation value. It does not takecare of the spreading of the wave function whichis accommodated by the squeezing of the wavepacket. For this, one introduces the metaplecticoperators.

Within the set of all unitary automorphismsof L2 (Rn), the metaplectic group Mp (2n,R)is the subgroup generated by the quadraticFourier transforms. One can show that thereexists a continuous group epimorphism, a pro-jection map, π : Mp → Sp that is the two–fold

covering of the symplectic group [47, 48]. Thismeans that for every symplectomorphism, i.e., alinear canonical transformation here, there ex-ist two associated metaplectic unitary operatorsthat differ by a sign. Then, only after specifyinga symplectic matrix, S, and a choice of a sign,σ = ±1, one can associate a metaplectic opera-tor, M(S, σ), that forms a group. In general, ametaplectic operator satisfies

M(S1)M(S2) = ±M(S1S2), (41)

M−1(S) = M †(S) = ±M(S−1). (42)

For our quadratic system, they propagate z asthe classical phase space vectors propagate, i.e.,

M †(S)zM(S) = Sz. (43)

Given a metaplectic operator associated withthe symplectic matrix S, there exists a corre-sponding unitary operator UM . In fact, thisoperator has been rediscovered by researchersin different fields many times and can be repre-sented in the position space as

UM (q, q) = 〈q(q)| M |q(q)〉

=σ

(2iπ~)n/2 √|detB|

exp

[i

2~

(q⊺DB−1q− 2q⊺B−1q + q⊺B−1Aq

)], (44)

so that this unitary operator acts as a propaga-tor for the wave function in position represen-tation, i.e.,

ψ (q) =

∫dq UM (q, q)ψ′ (q) , (45)

where the integral is taken from −∞ to +∞ asin other integrals that appear without indicatedlimits in this work. In order to understand theeffect of metaplectic operators acting on trans-lated states one uses the property

M †(S)T (z′) M(S) = T(S−1z′

), (46)

which is paramount for the investigation of thesqueezed coherent states.

From now on, let |0〉 represent the groundstate of a coherent state with 〈0 |z|0〉 = 0.Also consider a coherent state |z〉 with ex-pectation value 〈z〉 by keeping in mind that

|z〉 = T (z) |0〉. Then, the combined effect ofthe metaplectic and the Weyl–Heisenberg oper-ators on a coherent ground state is given by

M (S) T (z) |0〉 = T (Sz) M(S) |0〉= M(S) |z〉 . (47)

Next, it can be shown that there exists a welldefined unitary operator in L2 (Rn) that prop-

9

agates an initial state |z0〉 by

U (t, z0) = exp

(iγ(t)

~

)T (z(t)) M (S(t)) T † (z0) .

(48)

The propagator U (t, z0) accommodates the ac-tion of both the metaplectic and the translationoperators such that

|ψ〉 = U (t, z0) |ψ0〉 , (49)

for an arbitrary initial state |ψ0〉. Here,

γ (t) =1

2

∫ t

0

dt [pq− qp− 2H (z, t)] (50)

is a phase factor that does not necessarily ap-pear in all of the coherent state propagators inthe literature. As we follow Littlejohn’s argu-ment in [5] we choose to include this phase fac-tor so that the definition of translation opera-tors can be extended to have a group property.

In that case, it is not hard to show thatU (t, z0) satisfies the Schrodinger equation justlike the wave function itself,

i~dU

dt= HU , with U (0, z0) = Identity.(51)

In order to find out how this propagator actson an initial wave function which is centered atz0, one can make use of the ground state thatis centered at 0. Namely, substituting eq. (48)

in eq. (49) with |ψ0〉 = T (z0) |0〉 gives

|ψ〉 = exp

(iγ(t)

~

)T (z(t)) M (S(t)) |0〉 .(52)

Thus, regardless of what the initial state is,one can make use of a fiducial, ground statein order to obtain the final state. This mightseem counter intuitive at a first glance. How-ever, note that the information about the initialphase space vector is already included in thesymplectic matrix S and its quantum counter-part M(S).

Finally, since the ground state wave functionis represented in position space as

ψ|0〉 =1

(π~)n/4

exp

(−q⊺q

2~

), (53)

one can obtain the matrix representation of thesqueezed coherent state wave function definedat time t and centered at the phase space point〈z〉 = (〈q〉, 〈p〉)⊺ in its exact form as [5]

ψ =1

(π~)n/4

exp

(iγ(t)

~

)T (z(t))

1√det (A + iB)

exp

[i

2~(q⊺Γq)

]

=1

(π~)n/41√

det (A + iB)exp

[i

~

(γ(t) + 〈p〉⊺q− 〈p〉⊺〈q〉

2+

1

2(q− 〈q〉)⊺ Γ (q− 〈q〉)

)],

(54)

where Γ = (C + iD) (A + iB)−1.

Our aim in the current work is to find thecausal interpretation of a system represented bythe wave function ψ given in eq. (54). We willessentially study this system within the realmof statistical mechanics. Therefore, quantummechanical distribution functions are essentialfor our investigation. Thus, we will now intro-duce certain concepts that were previously in-

troduced into the literature and which will beuseful in our statistical construction.

D. Wigner function, Wigner ellipsoid and

the covariance matrix

For a given state |ψ〉, one can associate afunction, known as the Wigner function W =

10

W (q,p), which is the Weyl symbol of the pro-jection operator |ψ〉 〈ψ|. In the context of quan-tum mechanics, it was introduced into the lit-erature by Wigner [25] as a quasi–probabilitydistribution. Wigner function has similar prop-erties as the phase space distribution functionof classical mechanics which is preserved viathe Liouville equation. For a state, |ψ〉, in q–representation it is given by [25]

W =1

(2π~)n

∫ψ (x)ψ∗

(x˜)

exp

(− i

~px

)dx.

(55)

with

x = q +x

2and x

˜= q− x

2. (56)

Wigner function has very nice transforma-tion properties. Let us consider the Weyl–Heisenberg operator, T (z′), and the metaplec-

tic operator, M (S), introduced in the previoussection. It is known that if W (z) is the Wignerfunction of a state |ψ〉, then, W (z − z′) is the

one of the translated state T (z′) |ψ〉 [5]. In ad-dition, the transformation of the Wigner func-tion under a symplectic transformation givesW (S−1z). Namely, if W (z) is the Wigner func-tion of a state |ψ〉, then, W (S−1z) is the Wigner

function of the state M (S) |ψ〉. This results in

W (z, t) = W (S−1z, 0), (57)

such that the Wigner function is invariantthroughout the evolution of the system.

The expectation values of quantum operatorscan be obtained via the Wigner function with anintegral transform similar to the classical phasespace averaging. For an operator, F , for exam-ple, its expectation value is obtained through

〈ψ| F |ψ〉 =

∫dzW (z)f(z), (58)

where the phase space function f(z) is the Weyl

symbol corresponding to the operator F . Alsonote that the distribution W (z) is normalized,

i.e.,∫dzW (z) = 1. (59)

For a Wigner function, which is centered at〈z〉, the first order moments can be calculatedvia

〈z〉 =

∫dzW (z)z. (60)

Moreover, a covariance matrix, Σ, can be cal-culated in a similar manner via the second mo-ments of the Wigner function via

Σαβ =

∫dzW (z)zαzβ . (61)

Note that the matrix Σ is symmetric and non-negative. By using eq. (61), one can show forGaussian states that Wigner function in eq. (55)also takes a Gaussian form [5, 37, 49]

W = W (z, t)

=1

(π~)n exp

− 1

~(z− 〈z〉)⊺ W (z− 〈z〉)

,

(62)

where W = ~Σ−1/2.

We know that for the squeezed coherentstates, the first moments follow classical trajec-tories, i.e., 〈z〉(t) = S 〈z〉(0). Also the Wignerfunction being preserved in the phase spacegives

W(t) = S−⊺W(0)S−1, with detW = 1.(63)

Therefore, for the squeezed coherent states, Wis symplectic, symmetric and positive definite[49]. One can then choose, for example, [37]

(z− 〈z〉)⊺ .W. (z− 〈z〉) = 1, (64)

which defines the surface of an ellipsoid centeredat 〈z〉. This means that both the surface andthe center of the Wigner ellipsoid transformsrigidly throughout the evolution of the system[5, 37].

As the invariants of a system is associ-ated with its symmetries, one might wonder

11

which quantum invariant is preserved under thesymplectic symmetry of the classical system.In [37], Yeh shows that a generalized Lewis–Riesenfeld invariant operator1 can be definedfor a time dependent quadratic Hamiltonian fol-lowing the Weyl correspondence of the Wignerellipsoid. Let us consider a system associatedwith Gaussian states. Then, there exists a clas-sical invariant

I =1

2(z− 〈z〉)⊺ .W. (z− 〈z〉) , (69)

associated with the system that has a corre-sponding quantum invariant operator

I =1

2(z− 〈z〉)⊺ .W. (z− 〈z〉) , (70)

which satisfies dI/dt = 0 as in eq. (66). Thisfollows from the fact that not only the classicalphase space coordinates and momenta but alsotheir quantum correspondences evolve via linearsymplectomorphisms for quadratic Hamiltoni-ans. Then, the time evolution of z(t) = Sz(0)and 〈z〉 (t) = S 〈z〉 (0) cancels the time evolutionof the Wigner matrix, W(t) = S−⊺W(0)S−1,

1 Consider a Hamiltonian operator for a harmonic oscil-lator with a unit mass and time dependent frequency,Ω(t), in 1–dimension as

H (q, p; t) =1

2p2 + [Ω(t)]2q2. (65)

Lewis [28] and Lewis&Riesenfeld [29] define a dynamic

invariant operator, I = I(t), such that

˙I =

dI

dt=

∂I

∂t+

1

i~[I, H]. (66)

This invariant operator is given by [28, 29]

I =1

2

(

λq2

ζ2+

[

ζp− ζ q]2

)

(67)

provided that the complex variable ζ = ζ(t) satisfiesthe Ermakov equation

ζ + [Ω(t)]2ζ =λ

ζ3, (68)

where λ is a constant.

in the operator I above. Indeed, one can con-clude that for the squeezed coherent states ofthe quadratic Hamiltonians, invariance of thegeneralized Lewis–Riesenfeld operator followsfrom the correspondance between the invariantWigner ellipsoid and the density operator [37].

In this section, we presented the preliminar-ies which are essential for our investigation.Those seemingly unrelated ingredients come to-gether in the main body of our work in Sec-tion (III) in the following manner. We considera quadratic system with a classical symplecticphase space dynamics as in Section (II B). Aquantum correspondence of this system is con-sidered via Littlejohn’s squeezed coherent statewave function that was shortly presented in Sec-tion (II C). Note that we would like to inter-pret this system as in the de Broglie–Madelung–Bohm approach which was summarized in Sec-tion (II A). For our investigation, this is a hy-drodynamic interpretation intertwined with cer-tain statistical and thermodynamic concepts.Therefore, we will also make use of the defi-nition of the Wigner function and the Wigner–Weyl–Moyal correspondence as summarized inSection (II D).

III. A HYDRODYNAMIC

INTERPRETATION FOR GENERIC

SQUEEZED COHERENT STATES

Previously, in Section (II A), we summa-rized the causal approach of the de Broglie–Madelung–Bohm. Let us recall that the en-tire formalism depends on a wave function beingwritten on its polar form, i.e., ψ = R exp

(iS~

)

where R and S are real functions. Therefore,in order to start our investigation, we need totransform the wave equation of the squeezed co-herent state given in eq. (54) into its polar formfirst. This is what we present in the next sec-tion.

12

A. Foundations of the construction

1. Polar decomposition

The Hamiltonians we consider here are inquadratic order, thus any choice of operator or-dering will result in the same outcome. Let uschoose the symmetric ordering and consider thefollowing Hamiltonian operator

H =1

2q†aq +

1

2

(q†bp + p†b⊺q

)+

1

2p†cp.

(71)

Now we write the Schrodinger equation by usingthe generic Hamiltonian operator in eq. (71), sothat,

i~∂ψ

∂t=

(1

2q⊺aq− i~

2[Tr(b) + 2q⊺b∇q]

−~2

2∇

⊺

q c∇q

)ψ,

(72)

where

∇q =∂

∂q=

[∂

∂q1

∂

∂q2...

∂

∂qn

]⊺. (73)

Then, for a wave function which is written inits polar form (1), we obtain the pure imagi-

nary and the pure real parts of the Schrodingereq. (72) respectively as

∂R

∂t= −1

2

[R∇⊺

qc∇qS + (∇qS )⊺ c∇qR

+ (∇qR)⊺ c∇qS + 2q⊺b∇qR

+Tr(b)R] ,(74)

and,

∂S

∂t= −

[1

2(∇qS )

⊺c∇qS +

1

2q⊺aq

−~2

2

1

R∇

⊺

qc∇qR+ q⊺b∇qS

]. (75)

When we compare the real part of theSchrodinger equation in the generic case, i.e.,eq. (75), with the one of the original definitionin eq. (4), we realise that the general quantumpotential for an n–dimensional system is

Q = −~2

2

1

R∇

⊺

qc∇qR. (76)

This is the analogous expression for eq. (8).

In order to obtain the explicit form of theequation set (74)-(75) for generic squeezed co-herent states, we need to write the wave func-tion given in eq. (54), i.e.,

ψ =1

(π~)n/4

1√det (A + iB)

exp

[i

~

(γ(t) + 〈p〉⊺q− 〈p〉⊺〈q〉

2+

1

2(q− 〈q〉)⊺ Γ (q− 〈q〉)

)](77)

in its polar form. Note that it is notimmediately obvious whether this is possi-ble for a generic case due to the Γ =(C + iD) (A + iB)

−1term that appears in the

exponent and the√

det (A + iB) term that ap-pears in the denominator. However, we willsee that the modified Iwasawa factorization,eq. (28) of Section (II B), will help us in identify-ing the pure real and the pure imaginary partsof the wave function of the squeezed coherent

states.

Let us recall that the classical evolution of thesystem is governed by a symplectic matrix S asin eq. (23), whose action on the phase space canbe factored into three effects: shearing, magni-fication and rotation. The latter, the fractionalFourier transformer part of the Iwasawa factor-ization, is represented by a matrix f(u) given

13

by

f(u) =

[Reu Imu

−Imu Reu,

], (78)

as we have already represented in Section (II B).Here, u is a unitary matrix given by

u = (AA⊺ + BB⊺)−1/2

(A + iB) . (79)

Since unitary matrices can uniquely be writtenin the form u = uu where u ∈ SU(N) and u =exp (iα/n)In, we have detu = exp (iα), and

√det (A + iB) = exp (iα/2)

√det s. (80)

Note that s = (AA⊺ + BB⊺)1/2

= s⊺ is the ma-trix that is responsible for pure magnificationsin phase space as denoted in eq. (29). In that

case, we have2

Γ = (C + iD) (A + iB)−1

= (C + iD) (A− iB)⊺

[(A + iB) (A− iB)⊺]−1

= (CA⊺ + DB⊺ + iIn) s−2

= −g + is−2, (81)

where the third line in eq. (81) followsfrom the symplecticity conditions AB⊺ =BA⊺ and AD⊺ − BC⊺ = In of S givenin eqs. (24) and (26). Note that g =

− (CA⊺ + DB⊺) (AA⊺ + BB⊺)−1

= g⊺ ap-pears in the Iwasawa factorization (28) as partof the symplectic evolution that is responsiblefor the shearing effect in p–space. In optics, forexample, its associated symplectic matrix l(g)corresponds to the propagation through a thinlens.

Then, once we substitute the expressions(80) and (81) into eq. (77), we obtain the po-lar decomposition of the wave function of thesqueezed coherent state. We find the real am-plitude and the real phase function respectivelyas

R =1

(π~)n/4

1√dets

exp

[− 1

2~(q− 〈q〉)⊺ s−2 (q− 〈q〉)

], (82)

(83)

S = γ(t) − ~α

2+ 〈p〉⊺q− 〈p〉⊺〈q〉

2− 1

2(q− 〈q〉)⊺ g (q− 〈q〉) . (84)

It can clearly be seen that the real amplitude ofthe wave function is governed solely by the ma-trix s that appears in the symplectic magnifica-tion matrix m(s). On the other hand, the phasefunction is governed by the matrix g responsi-ble for the shearing/lensing effect, l(g), adopted

2 A similar result can be found in [50].

from the underlying classical phase space evo-lution.

In the next section, we will make use of thedefinitions reintroduced in Section (II D) in or-der to find a phase space distribution functionand a covariance matrix associated with thesqueezed coherent state wave function.

14

2. Wigner function and the covariance matrix

Recall that the Wigner function, being theWeyl symbol of the density operator, is a quasi–probability distribution of a phase space. Mean-ing, it might take negative values in general.However, for Gaussian states, Wigner functionis non–negative and thus, it can be used as aproper phase space distribution function in ourcurrent investigation.

In order to obtain the explicit form of theWigner function as presented in Section (II D),we follow [49] and [5] by considering the eq. (81).Then, for a Gaussian Wigner function,

W =1

(π~)nexp

[−1

~(z− 〈z〉)⊺W (z− 〈z〉)

],

(85)

the Wigner matrix, W, is a 2n× 2n symplecticmatrix that takes the form

W =

[ (s−2 + gs2g

)gs2)

s2g s2

], (86)

in our case.

As mentioned before, a covariance matrixcan be obtained via the Wigner distributionby making use of the eq. (61) such that Σ =~

2W−1. Then, we find the covariance matrix

associated with the squeezed coherent state as

Σ =

[σqq σqp

σpq σpp

]

=

⟨q2

⟩− 〈q〉2

⟨qp+pq

2

⟩− 〈q〉 〈p〉

⟨pq+qp

2

⟩− 〈p〉 〈q〉

⟨p2

⟩− 〈p〉2

=~

2

[s2 −s2g

−gs2(s−2 + gs2g

)].

(87)

Moreover, the invariance of the Wigner ellipsoiddictates that

Σ(t) = S(t)Σ(0)S⊺(t). (88)

Then, we get

dΣ

dt= LHΣ + ΣL

⊺

H, (89)

with

LH =

[b⊺(t) c(t)−a(t) −b(t)

], (90)

due to eq. (22).In order to find out how the sub–matrices

evolve, we substitute the explicit form of Σ ineq. (87), into its time evolution above. Then weobtain,

ds−2(t)

dt= −s−2b⊺ − bs−2 + s−2cg + gcs−2,

(91)

dg(t)

dt= a− bg − gb⊺ − s−2cs−2 + gcg.

(92)

We will make use of the equations (91) and (92)while studying the time evolution of hydrody-namic and thermodynamic variables in the fol-lowing sections.

Before moving on to a hydrodynamic inter-pretation, we will now have a consistency check.Those results will be very useful in analysing theenergy definitions presented in our work.

3. Consistency check: a pathway to

thermodynamics

In [5], Littlejohn argues that even thoughGaussian states have Gaussian Wigner func-tions, the converse is not necessarily true. Thismeans that the equality of two Wigner func-tions does not immediately imply the equalityof the corresponding wave functions. In orderfor a Gaussian wave function that is obtainedfrom a Wigner distribution to match the wavefunction obtained via the metaplectic operatorsacting on a ground state wave function, oneneeds to introduce a phase factor. Note thatthis phase factor is −α/2 which follows from theeigenvalues of the underlying fractional Fouriertransformer of the classical phase space evolu-

15

tion that we presented in eq. (80).

We will now have a consistency check whichmight seem redundant at a first glance. How-ever, during this process, we will obtain thevalue of α in terms of the elements of the Hamil-tonian matrix, LH, and the covariance matrix,Σ. This result will be crucially important whenwe introduce the hydrodynamic interpretationand the associated thermodynamic variables inthe following sections.

It is known that in order to obtain the wavefunction of the squeezed coherent states, onecan follow an alternative route than the one ofLittlejohn. For example, one method is to startwith an ansatz, such that the wave function isin the following form

Φ (q, t) = A (t) exp

(i

2~q⊺Γq

), (93)

during any point of the evolution. Here, Γ

belongs to the Siegel space of complex sym-metric matrices which is given as before, i.e.,Γ = (C + iD) (A + iB)

−1. Then, it is shown

that in order for Φ to be a solution of theSchrodinger equation, one needs to satisfy thefollowing two conditions [51]

dΓ

dt= −a− Γb⊺ − bΓ− ΓcΓ, (94)

dA

dt= −1

2Tr (b + cΓ) A , (95)

where A is given as A =

(π~)(−n/4)

(det [A + iB])−1/2

. Here, “Tr”represents the trace operator. Then, thecondition (94) of [51] corresponds to oureqs. (91) and (92) due to Γ being decomposedinto its pure real and pure imaginary compo-nents in eq. (81). Likewise, as we obtained√

det (A + iB) = exp (iα/2)√

det s previouslyin eq. (80), the condition (95) implies

dα(t)

dt= Tr

(cs−2

). (96)

We will elaborate on the importance of this re-sult in Section (III C 4).

B. Probability distributions

1. Phase space distribution and the entropy

Whether it is within the classical or withinthe quantum theory, there is no unique way ofapproaching the concept of probability distribu-tions and the entropy in general. For instance,as it is stressed many times in the literature,coherent states are the minimum uncertaintystates. Thus, they contain maximum informa-tion and minimum entropy. This statement isusually vaguely stated in the literature withoutspecifying in which manner the concept of infor-mation and entropy are defined. Here, we sug-gest certain definitions by considering the phasespace as our main object.

From now on, we will adopt the nomenclatureand the notation of statistical mechanics. Forexample, for a Gaussian probability distributionρ,

1

(2π)d/2

1√detM

exp

[−1

2(x− µ)⊺ M−1 (x− µ)

],

(97)

we will write

ρ := N (x|µ,M) , (98)

where x is a d–dimensional variable vector andµ is its mean with respect to the Gaussian dis-tribution, (97). The d×d matrix M is a positivesemi–definite covariance matrix of the distribu-tion. We will refer to our Gaussian Wigner func-tion as a joint probability distribution. This isdue to it including the information about both q

and p subspaces. We will denote this joint dis-tribution associated with the squeezed coherentstates, i.e., eq. (85), in the short form

ρ(q,p) := W (z) = N (z|〈z〉,Σ) , (99)

where 2n dimensional 〈z〉 are the standardexpectation values obtained by the averagingthrough the entire phase space, i.e., eq. (60).The covariance matrix Σ is given in its explicitform in eq. (87) for a squeezed coherent state.Note that the phase space probability distribu-

16

tion, ρ(q,p), is normalized.

We will now define a macroscopic joint en-

tropy, S(q,p), of the system which we write as

S(q,p) = −kB∫ρ(q,p) ln ρ(q,p)dqdp, (100)

by considering ρ(q,p) as in eq. (99). Computa-tion of the integral in eq. (100) now gives

S(q,p) =kB2

ln

(det

[2πe

~

2W−1

])

= kBn (1 + ln[π~ ]) +kB2

ln(det

[W−1

])︸ ︷︷ ︸

=0

,

(101)

which corresponds to an equilibrium entropy aswe have

dS(q,p)

dt= 0. (102)

The second term on the right hand side ofeq. (101) vanishes due to the Wigner matrix be-ing a symplectic matrix (and so is its inverse).Note that S(q,p) is indeed the Shannon entropy

of the Wigner function (multiplied by kB)3

which is sometimes referred to as the Wigner en-tropy [52–55]. It matches the Renyi−2 entropyup to a constant for Gaussian states [56]. It alsocorresponds to the lower bound of the missingposition and momentum information for a sta-tionary system as presented in [57] and whoserelation to the Heisenberg uncertainty principlehas been discussed in [58].

Indeed, the fact that S(q,p) corresponds to aminimum entropy state can be argued withinthe Schrodinger–Robinson uncertainty principle[59, 60]. Namely, the determinant of the covari-

3 Information entropy matches the thermodynamic en-tropy up to the factor kB for systems in equilibriumas in our case.

ance matrix, i.e.,

detΣ = det (σqq) det(σpp − σpqσ

−1qq

σqp

)

=

(~

2

)2n

(103)

corresponds to the minimum of theSchrodinger–Robinson uncertainty

σqqσpp ≥ σpqσqp +~2

4, (104)

which was originally defined for a 1–dimensionalconfiguration space. Thus, phase space entropytaking its minimum value is consistent with theminimum uncertainty and maximum informa-tion accommodated by the squeezed coherentstates. Note that for those states, if there ex-ists no classical phase space shearing/lensing,i.e., g = 0 in eq. (87), one has σpq = σ⊺

qp= 0

and the minimum of the standard Heisenberguncertainty is reached.

Moreover, S(q,p) being a minimum seemsto be consistent with certain topological argu-ments. Previously, it was recognized by de Gos-son [48, 61–63] that symplectic non-squeezingtheorem of Gromov [64] can be realized to de-fine some minimum uncertainty units on phasespace. Those are known as the quantum blobs.Namely, on the plane of conjugate canonicalpairs, there exist a minimum area of size π~.Due to the underlying symplectic capacity, thecanonical pairs that compose the projected areacan not take lower values. Here, we suggest thatS(q,p) reflects the missing information containedin the quantum blobs of de Gosson.

2. Marginal and conditional distributions

Even though phase space methods work sur-prisingly well, at least for the Gaussian states,it is the position space that we have imme-diate experimental access to. Naturally, thede Broglie–Madelung–Bohm theory was origi-nally presented in the q–representation. Thisrequires the introduction of marginal and con-ditional objects for the statistical considera-tions of quantum mechanics. For instance, in

17

[26], Moyal introduced space–conditional aver-

ages for the observables. Accordingly, Tak-abayashi argued that the quantum potential canbe considered as an apparent agent emergingfrom the configuration space projections of thephase space distributions [24].

Consequently, we introduce our marginal andconditional probability distributions now. Wewrite the marginal distribution as

ρ(q) =

∫ρ(q,p)dp, (105)

where ρ(q) is the q-space probability distribu-

tion whose value is equal to [R (q, t)]2. The

average of a function f = f(q,p) over themarginal distribution is obtained by

〈f〉(q) =

∫fρ(q)dq. (106)

However, ρ(q) includes only the informationthat is needed to describe the position coordi-nates. Once the positions are known, the re-maining, additional information needed in or-der to specify p is obtained by a conditionaldistribution which is sometimes referred to as aposterior distribution. We write the conditionalprobability distribution as

ρ(p|q) = ρ(q,p)/ρ(q). (107)

The average of a function f = f(q,p) over theconditional distribution can now be obtained by

〈f〉(p|q) =

∫fρ(p|q)dp. (108)

Once we compute the values of the marginal andthe conditional distributions for the squeezedcoherent states, we get

ρ(q) = N

(q|〈q〉(q),

~

2s2),

and

ρ(p|q) = N

(p|〈p〉(p|q),

~

2s−2

). (109)

Here,

〈q〉(q) = 〈q〉, and 〈p〉(p|q) = 〈p〉 − g (q− 〈q〉)(110)

are the mean values taken with respect to themarginal and the conditional distributions re-spectively. As we argued in Section (III B 1)Schrodinger–Robinson uncertainty principle ismore viable for generic squeezed coherent statesas for a generic case the minimum of the Heisen-berg uncertainty is not satisfied. At a firstglance, this seems to be contradicting with theidea of a minimum entropy state. However,Littlejohn attributes the minimum Heisenberguncertainty not being reached by the squeezedcoherent states to a geometric explanation [5].Specifically, it is due to the choice of a wrongsymplectic frame. He argues that it is onlywhen the principal axes of the Wigner ellip-soid coincide with the axes of positions and mo-menta, the minimum Heisenberg uncertainty isachieved. If they are not aligned, the angles ofprojections of the Wigner ellipsoid on the phasespace planes cause the system to appear as if itis not at a minimum uncertainty state.

Note that the original covariance matrix, Σ =~

2W−1, takes a block diagonal form

Σ =~

2

[s2 0

0 s−2

], (111)

when the lensing/shearing matrix satisfies g =0. This is when 〈p〉 = 〈p〉(p|q). Thus, our con-

ditional distribution ρ(p|q), which has a variance

matrix ~

2 s−2 for all cases, keeps track of the

minimum Heisenberg uncertainty with respectto conditional momenta.

Before going further into the thermodynamicinterpretation, we will now investigate more onthe probability distributions and their time evo-lutions.

18

3. The Fokker–Planck equation, probability fluxes

and the continuity equation

In statistical mechanics, the Fokker–Planckequation is considered as a stochastic differen-tial equation that gives the time evolution of aprobability distribution, ρ (x, t), and which fol-lows as

∂ρ

∂t= −∇xi

(βiρ) + ∇xi∇xj

(Dijρ) , (112)

where β (x, t) is the drift vector and D (x, t)is the diffusion matrix. It is known thatGaussian distributions are exact solutions ofthe Fokker–Planck equations. For a genericmulti–dimensional Gaussian distribution, ρ =N (x|µ,M) with M = M(t) as in eq. (98), thecorresponding drift vector and the diffusion ma-trix are

β(t) =dµ

dt, D(t) =

1

2

dM

dt. (113)

such that the Fokker–Planck equation can bewritten as

∂ρ

∂t=

[1

2(x− µ)

⊺M−⊺

dM

dtM−1 (x− µ)

+dµ

dt

⊺

M−1 (x− µ)

−1

2Tr

(dM

dtM−1

)]ρ.

(114)

Equation (114) holds for our joint distribu-tion, ρ(q,p) = N (z|〈z〉,Σ), our marginal

distribution ρ(q) = N

(q|〈q〉(q), ~2 s2

),

and our conditional distribution, ρ(p|q) =

N

(p|〈p〉(p|q), ~2 s−2

)once we replace the vari-

able vector, the mean vector and the covariancematrix in eq. (114) with the desired ones.

Let us now recall from the brief summary ofthe de Broglie–Madelung–Bohm approach givenin Section (II A) that the pure imaginary part ofthe Schrodinger equation can be interpreted as acontinuity equation, (6), given that the squared

amplitude of the wave function, [R (q, t)]2, is in-

terpreted as the density of a fluid. Within thestatistical interpretation, it gives the the prob-ability amplitude of a given outcome. Thus, itis no surprise that the Fokker–Planck equation,(114), of our marginal distribution, ρ(q), is inthe same footing as the imaginary part of theSchrodinger equation given in eq. (3), multipliedby R (q, t).

On the other hand, every probability distri-bution that satisfies a Fokker–Planck equationhas an associated probability flux that satisfiesa continuity equation. What we want to inves-tigate here is to see whether the Fokker–Planckinduced flux term is the same as the flux termthat appears in the hydrodynamic interpreta-tion of the Schrodinger equation.

Namely, for a generic Gaussian ρ =N (x|µ,M) which satisfies the Fokker–Planckequation, (114), one can define a probabilityflux

j = βρ−D∇xρ, (115)

with the drift vector and the diffusion matrixgiven in eq. (113) such that an associated con-tinuity equation

∂ρ

∂t+ ∇

⊺

xj = 0 (116)

is satisfied.

For our marginal distribution, for example,the continuity equation

∂ρ(q)∂t

+ ∇⊺

qj(q) = 0 (117)

is a Fokker–Planck equation with an associatedflux

j(q) =d〈q〉dt

ρ(q) −~

4

d(s2)

dt∇qρ(q)

=

[b〈q〉 + c〈p〉 +

1

2

d(s2)

dts−2 (q− 〈q〉)

]ρ(q).

(118)

This result follows from (i) the phase spaceexpectation values following classical trajecto-ries, i.e., d〈z〉/dt = LH〈z〉, (ii) ∇qρ(q) =

−2~−1s−2 (q− 〈q〉) ρ(q) being satisfied with

19

ρ(q) given in eq (109), and (ii)

d(s2)

dts−2 =

(b⊺s2 + s2b− cgs2 − s2gc

)s−2

(119)

via the evolution of the phase space covariancematrix, eq. (89), as indicated before.

Let us now compare the flux, j(q), in eq. (118)with the one of the de Broglie–Madelung–Bohmtheory, j, introduced in Section (II A). Notethat the latter is defined explicitly for a spe-cific Hamiltonian operator as in the Schrodingerequation (2). Moreover, as discussed in [65],its definition given by eq. (5) is non–unique.Also as discussed in [66] there exists an arbi-trariness on the definition a probability cur-rent in general. In order to make a connec-tion with the generic Hamiltonian here, we willtake 1/m → c where m refers to the massof the particle in the standard approach, asthe matrix c is responsible for the coupling ofthe momentum operator in our Hamiltonian ineq. (71). Also, as [R (q, t)]2 = ρ(q) we have

j = ρ(q)v = ρ(q)cp where v is considered asthe linear velocity. The term p = ∇qS issometimes referred to as the Bohm momentumand it is expected to satisfy the equation ofmotion (9). However, we emphasise that theoriginal de Broglie–Madelung–Bohm theory wasconstructed with such a choice of Hamiltonianoperator that the resultant Schrodinger equa-tion is interpreted within a hydrodynamic inter-pretation of irrotational fluid flow. That is whythe momentum p can be written as a divergenceof a potential/phase. We will discuss the actualmeaning of p in a short while. We should firstemphasize that our generic Hamiltonian oper-ator, (71), includes some position–momentumcoupling terms which are associated with therotational degrees of freedom in general. There-fore, the probability flux for the marginal distri-bution, j(q), includes the fluxes associated withthe rotations in addition to those associatedwith the linear motion.

In order to show this explicitly, let us assumein eq. (119), the symmetry of the products b⊺s2

and cgs2 for our symmetric matrices s, c and

g. Then, the flux associated with the marginaldistribution can be written as

j(q) = jirrot.(q) + jrot.(q) , (120)

with

jirrot.(q) = ρ(q)c〈p〉(p|q), (121)

and

jrot.(q) = ρ(q) [b〈q〉 + b⊺ (q− 〈q〉)] . (122)

Recall that the position–momentum couplingis represented by the matrix b in the genericHamiltonian. Therefore, jrot.(q) is the portion

of the marginal probability flux that includesonly those coupling terms. In the case thatb is symmetric, a velocity term can be associ-ated with rotational degrees of freedom, vrot. =jrot.(q) /ρ(q) = bq which takes a local form. This

is similar to a tangential velocity field for a ro-tational flow. Obviously, when b = 0, it is onlythe irrotational velocity virrot. = jirrot.(q) /ρ(q) =

c〈p〉(p|q) that governs the dynamics of the en-

semble. Here, 〈p〉(p|q) = 〈p〉 − g (q− 〈q〉) are

the conditionally averaged momenta given byeq. (110) previously. It can easily be checkedthat 〈p〉(p|q) = ∇qS holds by making use of

the phase function, S , eq. (84), of the squeezedcoherent state wave function. This means thatthe conditionally averaged momenta are equalto the Bohm’s momenta, p, and jirrot.(q) is equiv-

alent to j as expected.It is known that within a weak measurement

of Aharanov et al. [67], the post–selection ofpositions does not completely destroy the mo-mentum information and a mean, weak valueof momentum can be obtained for a system ofparticles. A weak value is in general a com-plex number and its real part is obtained by av-eraging the desired observable conditioned ona second measurement [68–70]. In this work,we consider momentum conditionally averagedon positions following Moyal’s [26] and Sonego’s[38] arguments. Also, it was previously realizedby many researchers that p are the real, mea-surable part of the weak value of the momenta[32, 71, 72]. Therefore, the conditionally aver-

20

aged momenta act as an effective momentumof a system. It is known that one can imag-ine a flow of an ensemble of particles such that〈p〉(p|q) = p represents a stream–line momen-

tum rather than the momenta of the individ-ual particles [38]. Our last remark is that it isthe matrix g, that is responsible for the classi-cal shearing/lensing effect, which differentiatesthe results of strong and weak measurementsof momenta for squeezed coherent states. Letus now compare our set up with some investi-gations in the literature that discuss stochas-tic quantum mechanics in relation to Einstein’swork on Brownian motion [73–75]. The impor-tant point we would like to start highlightinghere is that Einstein used some simplifying as-sumptions while building up his theory. Forsimplicity, he assumed the suspension of a par-ticle within a homogeneous and stationary liq-uid. Moreover, the time scale, over which thedynamics takes place is assumed to be smallerthan the observation time. In addition, randomdisplacements of the particle is assumed to besmall in order to give a time independent prob-ability distribution for the displacements. Only

then, his Fokker–Planck equation, which is alsoknown as the diffusion equation, takes its simpleform,

∂f

∂t= D∇2f. (123)

Here, the distribution function f = f(q, t) re-flects the number of particles per unit volumein 1-dimension, ∇ is taken with respect to theposition, and D is a constant diffusion coeffi-cient, contrary to our time dependent diffusionmatrices presented before. Then, the solution ofeq. (123) for the distribution function is givenby

f(q, t) =1√

4πDtexp

(− q2

4Dt

), (124)

with 〈q2〉 = 2Dt being the second moment ofthe displacements. Next, a drift effect can beadded by hand if there exists a constant externalforce acting on the particle in order to balancethe diffusion effect. This balance can be written

as

j = jdrift + jdiffusion = 0 → βf = D∇f

(125)

such that

∂f

∂t+ ∇j → ∂f

∂t= 0. (126)

In their formulation of quantum stochastic the-ory, Bohm and Hiley define a generic theorywhich, in its equilibrium, corresponds to stan-dard quantum mechanics [42, 76]. This isachieved by adding a stochastic contribution byhand to the standard probability flux j of thede Broglie–Madelung–Bohm theory. In order toform an analogous theory to Brownian motion,they refer to its version in Einstein’s work inwhich a particle is suspended in liquid under thegravitational force [77]. The osmotic/drift ve-locity and the diffusion flux they introduce arein the same form as they appear in Einstein’soriginal formulation. However, recall that Ein-stein’s work was constructed on a system re-stricted by some assumptions 4. Though, inBohm and Hiley’s work, the system does notnecessarily have to fall under such category.

Now, let us consider our marginal probabilityflux, j(q), given in eq. (118) which satisfies amore generic Fokker–Plack equation than theone of Einstein. An analogous balance equationas in eq. (125) is satisfied without introducingan extra flux term when

c (〈p〉 − g (q− 〈q〉)) = c〈p〉(p|q) = c∇qS = 0,

(127)

if b = 0 and cgs2 is symmetric. This refers to astationary state in which the system of particleshas zero average velocity. Thermodynamicallyspeaking, this should be reached at some abso-lute zero temperature. Note that such a result isconsistent with the quantum equilibrium condi-

4 In fact, those assumptions result in certain mathe-matical inconsistencies, including the breakdown ofGalilean invariance as discussed in [78].

21

tion defined in [42, 76] and it is obtained with-out using the simplifying assumptions of Ein-stein. Thus, it is a curious subject whether amore generic stochastic theory can be developedby considering relatively more generic Fokker–Planck equations as presented here.

The last point we would like to emphasise isthat even though the Fokker–Planck equationis usually associated with stochastic processes,it would be misleading to interpret the evolu-tion of the probability distributions of the gen-eralized squeezed coherent states investigatedhere in a stochastic manner. Note that thedrift vectors of the Gaussian distributions hereare obtained through the time evolution of theaverages of the corresponding phase space vec-tors. Those expectation values follow the samepath as the classical trajectories in phase space.Moreover, the associated diffusion matrices ofour distributions are obtained via the time evo-lution of the corresponding covariance matri-ces whose relation to the classical phase spacemagnifications and the classical shears has beenestablished in the previous sections. As thediffusion matrices of neither of the joint, themarginal, nor the conditional distributions ofours can be considered as random matrices,the evolution the corresponding probability dis-tributions can not simply be interpreted as astochastic process.

C. Thermodynamic variables

1. Back to entropy a la Sackur–Tetrode

Previously, in Section (III B 1), we defineda joint phase space distribution, ρ(q,p), andan equilibrium entropy, S(q,p). Later, in Sec-tion (III B 2) we defined a marginal distribution,ρ(q), for positions and a conditional distribu-tion, ρ(p|q), for momenta. Now we will definetheir associated entropies regarding an analogybetween the classical Sackur–Tetrode entropy[79, 80] of the kinetic theory, SST, and S(q,p).

For a system of particles, which is representedby the Boltzmann statistics, one can identifythe contributions to SST regarding the miss-

ing information of the particles’ continuous po-sitions and momenta as 5

SST(positions) = kBN lnV =

kBn

2lnL2,

SST(momenta) =

kBn

2ln (2πemT ). (128)

Here n = 3N for a number of N particles in3−dimensions. The cubical box that enclosesthe particles has a volume V = L3. The massof each particle is given by m and T is the tem-perature as it appears in the Boltzmann distri-bution.

One then includes certain corrections to theclassical SST. For example, by (i) consideringa finite, discretized space obtained via divid-ing the continuous space into (π~) sized boxesand neglecting the contribution, SST

(quantum) =

kBn ln (π~), coming from the quantum mechan-ical uncertainty within each quantum sized box,and (ii) subtracting the extra information inSST(positions) due to assuming that the particles

are distinguishable. Note that our phase spaceentropy, S(q,p), is equal to SST

(quantum) up to a

constant addition term. Now we would like todecompose S(q,p) in a fashion similar to the de-

composition of the classical part of the SST , i.e.,

SSTcl. = SST

(positions) + SST(momenta)

mS(q,p) = S(q) + S(p|q). (129)

For this, we define certain quantumentropies such that the entropy de-fined through the marginal distribution,

ρ(q) = N

(q|〈q〉(q), ~2 s2

), corresponds to the

missing information due to positions. We writeit as

S(q) = −kB∫ρ(q) ln ρ(q)dq. (130)

The entropy defined through our conditional

5 See, for example, the detailed discussions given in Sec-tion 4.3, Section 5.4 and Appendix L of [81].

22

distribution, ρ(p|q) = N

(p|〈p〉(p|q), ~2 s−2

)

corresponds to the missing information due tothe post–selected momenta and we write it as

S(p|q) = −kB∫ρ(q,p) ln ρ(p|q)dqdp. (131)

Then, we have

S(q) =kB2

ln

(det

[2πe

~

2s2])

, (132)

and

S(p|q) =kB2

ln

(det

[2πe

~

2s−2

]), (133)

hence

S(q,p) = S(q) + S(p|q) = kBn (1 + ln[π~ ]) .

(134)

Comparison of eqs. (128) and eqs. (130)-(131)shows that our marginal and conditional en-tropies are the analogues of SST

(positions) and

SST(momenta) within the quantum realm. The only

difference is that we treat positions and mo-menta on an equal footing. For instance, thedefinition of SST

(momenta) acknowledges the fact

that there exists a variance in momenta due tothe difference in the bulk motion of a systemand the classical peculiar velocities of the par-ticles. This is represented by the temperatureterm that appears in SST

(momenta). This is no dif-

ferent for our S(p|q) which represents the miss-

ing information due to the variance, ~

2 s−2, of

the conditional momenta. On the other hand,for the first case, since the system is composedof classical particles there is no spreading infor-mation within SST

(positions). While S(q) is defined

via the quantum mechanical position variance~

2 s2 as in the entropy for momenta.

The last point we would like to emphasiseis that whether it is a classical or a quantumdefinition, thermodynamic entropy is a macro-scopic object defined through the entire phasespace. If we want to talk about a thermody-namic equilibrium, it is the joint phase spaceentropy that defines an equilibrium state with

dS(q,p)/dt = 0. However, one can expect en-tropy production associated with the marginaland the conditional entropies. In fact, we have

dS(q)

dt= kBTr (b− gc) = −dS(p|q)

dt, (135)

due to eq. (91). This means that the informa-tion gain/loss in positions and in momenta areequal in magnitude and opposite in sign. Theycancel each other in order for the system to sat-isfy a dynamic thermodynamic equilibrium atall times.

2. Quantum pressure and quantum temperature

Now the question is whether or not the en-tropies we defined in the previous sections reallyfit into a thermodynamic picture within someanalogue quantum kinetic theory. In order toinvestigate this, we refer to Sonego’s work [38],in which he presents a detailed investigationof the hydrodynamic interpretation of quantummechanics for generic states.

Sonego considers the standard Hamiltonianas in the original de Broglie–Madelung–Bohmmethod. He starts his investigation by defininga pressure tensor that is written as

P = −~2

4ρ(q)c∇q∇

⊺

qρ(q) (136)

in our notation. By adopting some techniquesfrom the kinetic theory and by making use of theWigner function of the phase space, he showsthat the definition of the pressure tensor ineq. (136), indeed follows from a term

T (q, t) =1

kBc

∫ (p− 〈p〉(p|q)

)2

ρ(p|q)dp.

(137)

In that case, an equation of state P = TrP =ρ(q)kBT is satisfied with the temperature termT = TrT . The integral in eq. (137) essentiallygives the variance of momenta via which a tem-perature term is defined. This is similar tothe case of the classical kinetic theory. Whatis important here is that it is the variance of

23

the conditionally averaged momentum that de-fines a macroscopic phenomenon like tempera-ture here. This means weak measurements areagain at the center of the definitions of measur-able thermodynamic variables 6.

For the squeezed coherent states we havehere, the conditional distribution is ρ(p|q) =

N

(p|〈p〉(p|q), ~2 s−2

). Then, the variance of

conditional momentum is ~

2 s−2 and

kBT =~

2Tr

(cs−2

). (138)

Thus, we can write the conditional distributionin a Maxwellian manner as in the realm of theMaxwell–Boltzmann statistics, i.e.,

ρ(p|q) =1

(2πkB)n/2

1√det (c−1T )

exp

[− 1

2kB

(p− 〈p〉(p|q)

)⊺ (T

−1c) (

p− 〈p〉(p|q))]. (139)

3. Internal energy, quantum potential and a

conditional virial relation

Until now, we emphasized the importance ofthe measurements that are done with respectto the post–selection of positions. Let us nowdefine an energy that is obtained by coarse–graining the Hamiltonian operator of the sys-tem over the momentum variables. For this,we will make use of the conditional averages in-troduced in Section (III B 2). We write the re-sult as a functional of the classical Hamiltonian,H (q,p, t), i.e.,

U(p|q) (q, t) := 〈H〉(p|q)

= H(q, 〈p〉(p|q), t

)+

Ukin.︷ ︸︸ ︷1

2kBT .

(140)

Then, the conditional internal energy, U(p|q), iscomposed of (i) a portion including the classicalHamiltonian functional that inputs the effectivestreamline momentum of the flow of the systemas the momentum variable, (ii) a pure quan-tum contribution with an energy term, Ukin. =

6 Also see the discussion of [71] on the variance of theconditional momentum and its relation to weak mea-surements in the context of Sonego’s work.

kBT/2 analogous to the internal energy in clas-sical kinetic theory for a single degree of free-dom. In Sonego’s work [38] the term that cor-responds to our U(p|q) is referred to as a “lo-cal energy” due to its dependence on the localposition coordinates. Though, we should keepin mind that U(p|q) involves averages over themomentum variables which are already post–selected over positions.

Let us recall that in the de Broglie–Madelung–Bohm approach, it is the quantumpotential that is responsible for the observedquantum behaviour of a system. In eq. (76)of Section (III A 1), we derived it as Q =(−~2∇⊺

qc∇qR

)/2R for the higher dimensional

case. In order to find its value for a genericsqueezed coherent state, we substitute the realamplitude, R(q, t), given in eq. (82) into its def-inition. The result follows as

Q =~

2Tr

(c s−2

)− 1

2(q− 〈q〉)⊺ s−2c s−2 (q− 〈q〉) .

(141)

There is a common perception in the literaturethat Q→ 0 should hold as ~ → 0. This impres-sion follows from the fact that quantum poten-tial is the only term that distinguishes the clas-sical Hamilton–Jacobi equation from its quan-tum version. This is anticipated to be trueboth for the original derivation, eq. (4), and forour derivation for a generic quadratic system ineq. (75). Therefore, ~ → 0 is expected to give

24

the classical limit. However, such an expecta-tion does not hold for a standard coherent stateof a simple harmonic oscillator problem even inone dimensional, stationary case [34]. This dis-cussion is usually overlooked in the literatureexcept in a few studies. For example, in [82],this confusion is argued in detail. It is shownthat there exist different criteria to define a clas-sical limit, though they often seem to contradictwith each other. It is then resulted that certainstates do not have classical limits as ~ → 0 andthe authors suggest a method to properly definea classical limit [82]. We propose a different ex-planation here.

It is known that there are many interpreta-tions of the quantum potential for different sce-narios in the literature. Ours will be mostlyaligned with the ones in [39] and in [40] with im-portant differences. For example, the method-ology and the set up of [39] is very different thanours. However, the provided link between thequantum potential and the kinetic internal en-ergy of the system is quite similar. Let us nowdiscuss [40], in which the authors interpret Q asthe internal energy of a system for stationarystates. For example, for a 3–dimensional simpleharmonic oscillator, with frequency ω being thesame for each degree of freedom, they obtain

Qω =3~ω

2− 1

2mω2|r|2, (142)

where r is the position vector and m is the mass.Their interpretation is that the internal energyis given by the ground state energy, 3~ω/2 mi-nus the potential energy. However, the problemof ~ not appearing in the so–called potentialenergy term brings us to the discussion of theprevious paragraph. Can the two terms thatappear on the right hand side of eq. (142) be

treated equally? Besides, the internal energy isknown to be a coarse–grained object in thermo-dynamics. Thus, the value of Q itself can not beexpected to give the internal energy. Accordingto us, it is rather the expectation value ofQ thatshould be interpreted as the internal energy.

As the quantum potential depends on the po-sitions and on time only, 〈Q〉(p|q) = Q and

〈Q〉 = 〈Q〉(q) =~

4Tr

(c s−2

). (143)

Thus, following the value of temperature givenin eq. (138), we obtain

〈Q〉 =kBT

2= Ukin.. (144)

This gives exactly the quantum mechanical in-ternal energy in the form that it appears in thekinetic theory. Moreover, physically meaning-ful, measurable quantities are given by the av-erage values of the operators. Thus one shouldexpect 〈Q〉 → 0 as ~ → 0, which is the casehere.

We also observe that the maximum value ofthe quantum potential is obtained at q = 〈q〉.This is where the Gaussian position distribu-tion ρ(q) also peaks. Moreover, Qmax. = 2〈Q〉.This was found “interesting” without furtherexplanation in [83] in which the quantum po-tential and its mean are obtained for the Gaus-sian states. In fact, Q in eq. (141) signals anobject which is expanded around its maximumvalue. In classical thermodynamics, such ex-tensions are usually introduced to study fluc-tuations of thermodynamic variables at equilib-rium [84].

Let us now calculate the variance of the

quantum potential,⟨

(∆Q)2⟩

=⟨

(Q− 〈Q〉)2⟩

around its maximum value, i.e.,

25

⟨(∆Q)2

⟩=

⟨(dQ

dq

∣∣∣q=〈q〉

(q− 〈q〉) +d2Q

dq2

∣∣∣q=〈q〉

(q− 〈q〉)2)2

⟩(145)

=(s−2cs−2

)ij

(s−2cs−2

)kn

⟨(q− 〈q〉)i (q− 〈q〉)j (q− 〈q〉)k (q− 〈q〉)n

⟩(146)

=~2

4

(s−2cs−2

)ij

(s−2cs−2

)kn

(s2ijs

2kn + s2iks

2jn + s2ins

2jk

)(147)

= k2B

([TrT ]2 + 2TrT 2

)= 3k2BT

2 − 4k2B∑

i<j

λiλj . (148)

Here, the second line follows from the fact thatthe first order fluctuation term vanishes at q =〈q〉. The third line follows from the fact thatthe higher order moments of a Gaussian dis-tribution can be written as a function of thevariance due to the Isserlis Theorem [85]. Here,the variance in question is σqq = ~

2 s2. The last

line follows from the temperature matrix givenin eq. (137) with λi being its eigenvalues. Thevariance of quantum potential is thus propor-tional to the square of the temperature similarto the mean–square fluctuation of the energyin classical thermodynamics. Thus, we suggestthat Q involves both the kinetic internal energyat equilibrium and the fluctuations around it.

On the other hand, it is also known that for asystem in thermodynamic equilibrium, a virialrelation is satisfied if the system concurrentlysatisfies a hydrodynamic equilibrium. We basedour thermodynamic equilibrium on the invari-ance of our phase space entropy. In addition,our system satisfies a hydrodynamic equilib-rium due to the probability distributions sat-isfying the Fokker–Plank equations as given inSection (III B 3). Note that the standard virialrelation in quantum mechanics has two typi-cal derivations: (i) the commutator proof whichmakes use of the invariance of the infinitesimalgenerator i (qp + pq) /2 and is found in manytextbooks (cf. [86]) that goes back to Finkel-stein [87]; (ii) the proof which makes use of thegroup of dilatations that was first given by Fock[88]. In the literature, the problems regard-ing both of these methods have been investi-gated in many studies and alternative deriva-tions have been proposed [89–93]. Our aim

here, on the other hand, is to suggest a coarse–grained version of the virial relation applicablefor the statistical hydrodynamic interpretationwe presented here.

Consider the quadratic system we have thatis in thermodynamic equilibrium with its sur-roundings. Accordingly, we define a quantumvirial relation as

2Ukin. = −Upot. := −〈(q− 〈q〉)⊺ F 〉 ,(149)

where F = ∇qQ is the quantum force. Equa-tion (149) is different from its standard ana-logue in the sense that it represents an effectivesystem. For example, the temperature and thusthe Ukin. term exist solely due to the conditionalvariance of the momenta. We must also addthat the virial relation above incorporates solelythe quantum effects by considering a potentialenergy term derived from the quantum poten-tial only. One could in principle consider a moregeneric form of this virial relation which incor-porates a classical kinetic energy and a classicalpotential in addition to their pure quantum ana-logues. We believe that for such an investigationone should consider a mixed quantum–classicalphase space formalism, for example, as in [94].

While closing this section we should empha-sise that up until now we investigated the en-ergy coarse–grained over momenta. In order toobtain the internal energy coarse–grained overthe entire phase space, one needs to consider theexpectation value of the Hamiltonian operatorobtained through the joint phase space distri-bution. Then, we obtain a phase space internal

26

energy as

U(q,p) : = 〈H〉

= H (〈q〉, 〈p〉, t) + Ukin. +~

4Tr

(as2

)

− ~

4Tr

(2bgs2 − cgs2g

). (150)

Here, again, the first term on the r.h.s isthe classical contribution to the energy regard-ing the Hamiltonian functional that inputs thephase space average of the positions and mo-menta. The second term is the quantum ki-netic internal energy on account of the vari-ance of the conditional momenta. Recall thatif the shearing/lensing matrix, g, is zero, then〈p〉(p|q) = 〈p〉. In that case, the only term that

differentiates the global energy, U(q,p), from aconditionally averaged local one, U(p|q), is the

third term, ~Tr(as2

)/4, that is the energy con-

tribution coming from the variance of positions.This term has no analogue in classical theory asone assumes no variance in classical positions ingeneral, at least theoretically.

4. Relationship between the kinetic internal

energy and the Maslov index

A more profound observation is the relation-ship of the conditional kinetic internal energy,Ukin., to the Maslov index. The latter, de-noted by µ, is an (half)–integer valued map thatis usually associated with the closed loops ofthe Lagrangian subspaces of a symplectic vectorspace [95]. It is also interpreted as a topolog-ical invariant which gives the winding numberfor periodic systems [96]. For example, for aharmonic oscillator in 1-dimension,

∮pdq = EnT = 2π~

(n+

µ

4

), (151)

where En is the energy of the nth energy level,T is the period of the oscillations and µ takesthe value of 2.

Here, we will refer to an extended defini-tion of the Maslov’s formula for generic sym-plectic paths defined by linear symplectomor-

phisms [47, 97, 98]. In that case the symplecticphase space transformation matrix of the sys-tem is also periodic. Suppose that there ex-ists a 2n × 2n matrix f(u) which is both or-thogonal and symplectic. Then f(u) has an n–dimensional unitary representation, u = x+ iy,with real x and y, i.e., u(n) ∼ Sp(2n)∩O(2n).One then defines a Lagrangian subspace, Λ ∈L (n), via

Λ = Im

(x

y

). (152)

Now consider a loop Λ(t) = Λ(t + T ) ∈ L (n),where T = 1 is the normalized period of thesystem. Then a Maslov index can be definedfor this loop as [98]

µ = α(1) − α(0), (153)

where

eiπα(t) = detC [u(t)] , u(t) = (x(t) + iy(t)) ,

Λ(t) = Im

(x(t)y(t)

), ∀ t ∈ R.

(154)

Thus, for a periodic linear system whose phasespace transformations are governed by an or-thogonal symplectic matrix, f(u), it is thechange in the argument of the determinant ofthe corresponding unitary matrix, u, that de-fines the Maslov index.

In this paper, we consider systems that arenot necessarily periodic in general. Therefore,the evolution of the expectation values of theposition and momentum operators are governedby those symplectic matrices which are not nec-essarily orthogonal and periodic. In general,the energy is not a constant of time that candirectly be related to the Maslov index. How-ever, recall that in Section (II B), we applied anIwasawa factorization, eq. (28), to the govern-ing symplectic matrix, S. Its fractional Fouriertransformer component is an orthogonal sym-plectic matrix and denoted by f(u). This ma-trix is the portion responsible for the genericrotations in phase space which are not neces-sarily around closed loops. Note that the ma-

27

trix f(u) has a unitary representation, u. Then,α(t) = arg (detu) manifested itself in the phasefunction, S , in eq.(84), when we derived thegeneric squeezed coherent state wave functionin its polar form. Later on, in Section (III A 3),we provided a consistency check which seemedredundant earlier on. Namely, we showed thatin order for the squeezed coherent state wavefunction obtained from a Wigner distributionto uniquely match a wave function which is ob-tained through the action of the metaplectic op-erators on a ground state (and which preservesits form as in eq. (93) throughout the evolution),the condition dα(t)/dt = Tr

(cs−2

)has to be

satisfied. Subsequently, the rate of change of theargument of the unitary matrix u is Tr

(cs−2

)

which is defined in a similar fashion as the origi-nal Maslov index given in eq. (153). Then, eventhough the system is not necessarily periodic,the change in α is a measure of the kinetic inter-nal energy of the system, Ukin. = ~Tr

(cs−2

)/4.

This means that the generic phase space ro-tations, i.e., fractional Fourier transformations,are directly related to the quantum energy con-tent of the semiclassical systems in any case.

IV. SUMMARY AND CONCLUSION

In our investigation, we studied squeezed co-herent states in n–dimensions which are moregeneric than the standard ones. Those statescan be exactly defined for systems with timedependent Hamiltonians. Moreover, the min-imum uncertainty principle still holds at alltimes, without the uncertainty in positions andin momenta being necessarily equal. Squeezedcoherent states are mostly relevant for semi–classical physics or systems whose Hamiltonianis in quadratic order with respect to the phasespace coordinates.

Within different interpretations of quantummechanics, we chose to analyze the squeezedcoherent states within a hydrodynamic inter-pretation which allowed us to investigate thesystem thermodynamically. While doing thatwe placed the Wigner–Moyal–Weyl correspon-dence at the core of our construction as in [5]

where Littlejohn investigates the squeezed co-herent states in detail. Eventually, we out-lined a wide perspective by providing variouslinks between the classical and the quantum me-chanical paths in addition to highlighting cer-tain statistical concepts that are mostly rele-vant for our hydrodynamic and thermodynamicanalysis. We summarize those conceptual linksschematically in FIG (1).

It is known that dynamics of a linear clas-sical system is represented by linear symplectictransformations in a classical phase space. Sucha system is driven by a quadratic Hamiltonianwhose quantum analogue is also a quadraticfunction of position and momentum operators.In [5], Littlejohn derives the exact Gaussianform of the wave function of the squeezed co-herent states by making use of: (i) the Weyl–Heisenberg operators which translate a state inthe phase space (ii) the correspondance betweenthe symplectic group and the metaplectic oper-ators, the latter of which provide the spreadingof the wave function.

In order to investigate the squeezed coherentstates within the hydrodynamic interpretation,we started with decomposing the correspond-ing wave function into its polar form as in thede Broglie–Madelung–Bohm approach. In themean time, a phase space probability distribu-tion was obtained via the Wigner function as in[5]. Note that as our wave function in questionis Gaussian, the associated Wigner function isnon–negative and also Gaussian. This allowsit to be a proper candidate for a phase spacedistribution function.

Next, we started the thermodynamic analysisby defining a Shannon entropy via the Wignerfunction. For the case of the squeezed coher-ent states, this phase space Wigner entropy isa constant of time. Indeed, it taking a mini-mum value is consistent with the minimum un-certainty principle satisfied by the squeezed co-herent states. This is why we claimed that thesystem in question is in a dynamic thermody-namic equilibrium.

Further thermodynamic analysis was slightlymore involved as it requires one to incorpo-rate the statistical concepts with the quantum

28

mechanical measurement process. For exam-ple, the hydrodynamic interpretation we fol-lowed was derived within the position represen-tation. Momenta of the particles, on the otherhand, are identified once the positions are se-lected. Therefore, if we would like to decomposea phase space distribution function into its por-tions involving the information about the posi-tions and the momenta, we need to acknowledgethe fact that the momenta are post–selected.Indeed, this fact was realized even in the earlytimes of the statistical interpretation of quan-tum mechanics by Moyal [26] and by Takabayasi[24].

Therefore, we treated the phase space distri-bution function associated with the squeezedcoherent states as a joint distribution and de-composed it into two portions: (i) a marginaldistribution for positions which is equal to thesquared real amplitude of the wave function; (ii)a conditional distribution for momenta which isconditioned on positions. Note that our distri-bution functions are in Gaussian form and theyall satisfy the Fokker–Planck equation exactlywhich we also discussed in detail. Moreover,we showed that the conditionally averaged mo-mentum is equal to the so–called Bohm’s mo-mentum whose physical interpretation is stillunder debate. According to the hydrodynamicinterpretation, Bohm’s momentum reflects thestreamline momentum of the particles that con-stitute the system. It being equal to the condi-tionally averaged momentum also shows that itis an effective object. This was also realized incertain other studies [38] and its relation to theweak measurements of Aharonov et al. [67] hasbeen established before [32, 71, 72].

Next, we returned back to the thermody-namic analysis via decomposing the phase spaceentropy into two portions by making use of: (i)the missing information contained in the posi-tions which is obtained through the Shannon en-tropy of the marginal distribution; (ii) the miss-ing information contained in the post–selectedmomenta which is obtained through the Shan-non entropy of the conditional distribution ofmomenta. This is indeed analogous to the de-composition of the classical part of the Sackur–

Tetrode entropy of the kinetic theory as dis-cussed in the relevant section.

After defining the probability distributionsand their associated entropies, we followedSonego’s work [38] in order to define a quan-tum pressure and a quantum temperature forthe squeezed coherent states. Those satisfy anequation of state as in the classical kinetic the-ory. In the classical case, temperature is de-fined via the variance of the momentum whichis sourced by the peculiar velocities of the parti-cles with respect to the ensemble. Here, we ex-plicitly showed that the quantum temperatureis defined via the variance of the conditional mo-mentum distribution in a similar fashion. Thisallowed us to rewrite our conditional distribu-tion in a Maxwellian manner as in the Maxwell–Boltzmann distribution of the classical kinetictheory.

We further associated a conditional internalenergy to our system in equilibrium. This con-ditional energy includes a contribution comingfrom the classical Hamiltonian which is modi-fied by the conditionally averaged momenta anda portion that includes the quantum tempera-ture term as in the form of the internal energyof the classical kinetic theory. Later on, wedemonstrated the relationship of this kinetic in-ternal energy term with the quantum potentialof the de Broglie–Madelung–Bohm theory. Ac-cording to us, the quantum potential includesthe internal kinetic energy and its fluctuationsaround it at equilibrium. Accordingly, it is theexpectation value of the quantum potential thatgives the internal kinetic energy of the system.Eventually, we suggested a conditional virial re-lation that associates the kinetic internal en-ergy of the system with a potential energy termsourced solely by the quantum potential.

In brief, our outcome is a quantum kinetictheory associated with n–dimensional squeezedcoherent states in which the underlying ther-modynamics is time dependent. This is unlikeother works in the literature where the mainidea behind the construction of quantum ther-modynamics is usually adopted from the stan-dard classical thermodynamics. Namely, thesystem is assumed to relax to an equilibrium

29

in time and the energy of the system is keptconstant throughout the evolution. We believethe lack of time dependent quantum thermody-namic investigations follows from the lack of atime dependent equilibrium classical thermody-namics formalism constructed on a symplecticphase space.

Certain delicate issues also caught our at-tention throughout our investigation. Thefirst one follows from the Iwasawa factoriza-tion of the symplectic transformation matrixof the underlying classical system. This fac-torization essentially allows one to identify thelensing/shearing, pure magnification and therotation–like portions of the phase space trans-formations. For the squeezed coherent states,those sub–transformations play essential rolesin the quantum picture. For example, thesquare of the magnifications in classical posi-tions manifests itself as a quantum variance ofthe position uncertainty. Likewise, the square ofthe demagnification in classical momenta man-ifests itself as the quantum variance of the con-ditional momentum uncertainty. The matrixthat is responsible for a shearing effect in theclassical phase space also appears in the phasespace covariance matrix. In addition, it is thisshearing term that differentiates the conditionalaveraged momenta from the phase space aver-aged ones. Recall that the former are equalto the real part of the weak measurements ofmomenta. To be more specific, when the un-derlying classical trajectory has zero shearingeffect, the conditionally averaged momenta, orBohm’s momenta, are equal to the phase spaceexpectation values of the momentum operator.Then, the covariance matrix takes a block diag-onal form and one satisfies an exact Heisenberguncertainty.

On the other hand, unlike the shears and themagnifications, it is the classical rotation–liketransformation that manifests itself in the quan-tum kinetic internal energy of the system. To bemore specific, there exists a unitary matrix rep-resentation of the fractional Fourier transformerpart of the symplectic matrix that guides theclassical evolution. It is the time rate of changeof the argument of the determinant of this uni-

tary matrix which tell us about the quantumkinetic energy content of a system. This is sim-ilar to the definition of the Maslov index char-acterized for the symplectic paths. The Maslovindex also contributes to the energy content ofa system with periodic orbits.

As we indicated before, we considered onlythe squeezed coherent states of a linear systemhere. If we were to include the thermal statesto our investigation, then we could still definea Gaussian Wigner function and a phase spaceprobability distribution. However, in that case,the Wigner matrix in question would not be asymplectic matrix and its evolution in phasespace would not be so trivial (cf. [37]). Thiswould result in a Shannon entropy of the phasespace distribution which is not a constant oftime, meaning the system would not be at itsequilibrium. We believe this is a good pointto start investigating the non–equilibrium ther-modynamics of linear systems. In that case,the definition of the Wigner ellipsoid, the cor-responding information entropy and the Maha-lanobis distance of statistics [99] seem to be in-terconnected. We leave these issues for our fu-ture project.

Finally, in this work, we stayed within thelinear regime only. Even though this can beseen somewhat restrictive, it is still relevantfor certain application areas within the fieldsof quantum thermodynamics and quantum en-gines. Specifically, our results might find somearea of use within the quantum optomechanicalproblems as in [100–105] where stability and ef-ficiency issues are open problems for time de-pendent systems.

30

FIG. 1. The conceptual relations relevant for the hydrodynamic interpretation of the generic squeezedcoherent states outlined in this work.

ACKNOWLEDGMENTS

The author thanks Thomas Buchert, Prze-mys law Ma lkiewicz and Jan Jakub Ostrowski

for their comments. This work is a part ofa project that has received funding from theEuropean Research Council (ERC) under theEuropean Union’s Horizon 2020 research and

31

innovation programme (grant agreement ERC advanced Grant 740021ARTHUS, PI: ThomasBuchert).

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