The Dynamics of Compact Laser Pulses

0.55 MB·English

by  S. Goto

The Dynamics of Compact Laser Pulses
S. Goto1, , R. W. Tucker2 and T. J. Walton3
∗
1 Institute for Molecular Science, 38 Nishigo-Naka, Myodaiji, Okazaki 444-8585, Japan
2 Department of Physics, University of Lancaster and Cockcroft Institute, Daresbury
6 Laboratory, Warrington, UK
1 3 Department of Mathematics, University of Bolton, Deane Campus, Bolton, UK
0
2

y
a Email: [email protected], [email protected], [email protected]
M

3
Abstract

]
h
We discuss the use of a class of exact finite energy solutions to the vacuum
p
- source-free Maxwell equations as models for multi- and single cycle laser pulses
t
n in classical interaction with relativistic charged point particles. These compact
a
solutions are classified in terms of their chiral content and their influence on
u
q particularchargeconfigurationsinspace. Theresultsofsuchclassicalinteractions
[ motivate a phenomenological quantum description of a propagating laser pulse in

4 a medium in terms of an effective quantum Hamiltonian.
v
3
5
7
7 PACS numbers: 03.50.De, 42.50.Ct, 42.55.Ah,
0
.
1
0
5
1
:
v 1 Introduction
i
X
r Advancesinlasertechnologyhavemadepossibletheexplorationofphysicalprocesseson
a
unprecedented temporal and spatial scales. They have also opened up new possibilities
for accelerating charged particles using laser-matter interactions. Multi- and single cycle
high intensity (1010 1015Wcm 2) laser pulses can be produced using Q-switching or
−
−
mode-locking techniques [1]. Pulses of even higher intensity ( 1021Wcm 2) could
−
∼
accelerate charged particles such as electrons to relativistic speeds where radiation
reaction and quantum effects may influence their dynamics [2]. Lower intensity pulses
have also been used as diagnostic tools for exploring the structure of plasmas in various
Currentaddress: DepartmentofAppliedPhysicsandMathematics,GraduateSchoolofInformatics,
∗
Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japan
1
states [3,4]. In order to interpret experimental data involving classical laser interactions
with both charged and neutral matter, theoretical models [5–8] rely crucially on
parameterizations of the electromagnetic fields in laser pulses, particularly in situations
where traditional formulations using monochromatic or paraxial-beam approximations
have limitations [9–11]. Such theoretical models may not be compact in all spatial
dimensions and the role of a laser pulse as a classical probe is further limited by the
scales that it is designed to resolve. Pulse shape design characteristics are often guided
by simulations of laser-matter interactions which incorporate the known laws of physics
of relevance at such scales. For example, the intense experimental activity currently
exploring the electromagnetic properties of single-cycle laser pulses with nanoscale
objects (such as dielectric and plasmonic nanoparticles) demands efficient modelling
tools that accommodate the spatial compactness of such pulses. Such tools will
eventually require incorporation of quantum effects associated with these interactions
in order to properly describe observations and yield practical applications involving
nanostructured dielectrics and plasmonic metals [12]. Furthermore an effective quantum
description of the laser pulse itself would offer a new simulation tool for designing more
accurate methods of encoding quantum information.
The quanta associated with plane time-harmonic electromagnetic fields in vacuo provide
ideal discrete two-state systems (photons) that are routinely used as controllable qubits
in information science. Three-level bosonic quantum systems composed of two photons
in the same spatial and temporal configuration have also been contemplated [13] in
attempts to construct more efficient quantum gates for quantum communication. If
one regards a free propagating classical single-cycle (therefore non-time-harmonic) laser
pulseinvacuoasaspatiallycompactclassicalelectromagneticconfigurationwithdefinite
energy and temporal width t , one expects that when t (cid:46) (cid:126) its dynamical evolution,
0 0
E E
in both vacuo and material media, should be controlled by an effective quantum
Hamiltonian, rather than a classical one as is done in [14]. Such quantized collective
states could then be entangled with other quantized pulses or free photon states and
their interaction with classical or quantized states of electrically neutral continua (e.g.
optically inhomogeneous and anisotropic dielectrics or plasmas) or charged matter (e.g.
trapped ions [13]) may be worthy of investigation for technological applications such as
quantum computing and encryption.
In this article we first discuss a viable methodology for parameterizing a particular class
of propagating solutions to the source-free classical Maxwell equations in vacuo that
offers an efficient means to explore the classical effects of compact laser pulses on free
electrons in dynamical regimes where quantum effects are absent. The parameterization
is constructed from a remarkable class of explicit solutions of the scalar wave equation
found by Ziolkowski [15–19] following pioneering work by Brittingham [20] and Synge
[21]. Solutions in this class are parameterized in terms of three real constants that
are sufficient to completely determine the characteristics of any freely propagating
laser pulse in full accord with Maxwell’s equations in free space. They describe
2
solutions with finite total electromagnetic energy, electromagnetic fields bounded in
all three spatial directions and experimentally distinguishable chiral configurations.
With simple analytic structures their diffractive properties can be readily calculated
together with the behaviour of relativistic charged particle-pulse interactions over a
broad parameter range without recourse to expensive numerical computation. Based
on this behaviour and by analogy with the effective Hamiltonian theory of diatomic
molecules, we are led to construct a Hilbert space on which to describe certain quantum
states ofanelectromagneticpulseundergoingunitaryevolutiongeneratedbyaneffective
phenomenological quantum Hamiltonian in a medium. Such a Hamiltonian, defined in
termsofasetofparametersassociatedwithitsmediuminteraction, servesasaneffective
quantum model for the quantum pulse evolution.
2 Parameterizing Compact Electromagnetic Laser
Pulses
If a complex scalar field α satisfies (cid:3)α = 0 and Π is any covariantly constant (degree
µν
2) anti-symmetric tensor field on spacetime (i.e. Π = 0 for all µ,ν,δ = 0,1,2,3), then
µν;δ
the complex tensor field F = ∂ A ∂ A satisfies the source-free Maxwell equations
µν µ ν ν µ
−
in vacuo with:
(cid:112)
A = ∂ (α Π ) (cid:15)γµβ g (1)
ν γ µβ ν
| |
where g is the modulus of the determinant of the spacetime metric tensor field g with
| |
components g and (cid:15)γµβ denotes the Levi-Civita alternating symbol. In the following,
µν ν
we restrict to Minkowski spacetime, in which case the components Π can be used to
µν
encode three independent Hertz vector fields and their duals .
†
General solutions to (cid:3)α = 0 can be constructed by Fourier analysis. In cylindrical polar
Minkowski coordinates t,r,z,θ , axially symmetric solutions propagating along the z-
{ } (cid:82)
axishave, forz 0, thedoubleintegralrepresentationα(t,r,z) = ∞ dωe−iωtα(cid:101)(ω,r,z)
≥
where: −∞
(cid:90) ω (cid:90)
α(cid:101)(ω,r,z) = c kfω(k)J0(kr)e±iz√(ωc)2−k2dk + ∞kfω(k)J0(kr)e−z√k2−(ωc)2dk
0 ω
c
in terms of the zero order Bessel function and the speed of light in vacuo c.
Conditions on the Fourier amplitudes f (k) can be given so that the Hertz procedure
ω
above gives rise to real singularity free electromagnetic fields with finite total
electromagnetic energy. A particularly simple class of pulses that can be generated
In the language of differential forms on Minkowski spacetime A = (cid:63)d(αΠ), F = dA where
†
d(cid:63)dα=0, the 2-form Π satisfies Π=0 and (cid:63) denotes the Hodge map associated with g.
∇
3
P
e
fil
o
r
p 4
r
e
w 2
o
P
0
4
− −2 0 2 −2 Y 104
4 4 ×
X 104 −
×
Figure 1: Power profile P of the (CM,1) laser pulse at Z = 0, T = 0 with parameters
Λ = 600,Ψ = 1,Ψ = 1000,Φ = 0.001,Ξ = 1
1 2
{ }
in this way follows from the complex axi-symmetric scalar solution:
(cid:96)2
α(t,r,z) = 0 (2)
r2 +(ψ +i(z ct))(ψ i(z +ct))
1 2
− −
where (cid:96) ,ψ ,ψ are strictly positive (real) parameters with physical dimensions of
0 1 2
length. The relative sizes of ψ and ψ determine both the direction of propagation
1 2
along the z axis of the dominant maximum of the pulse profile. When ψ ψ ,
1 2
− (cid:29)
the dominant maximum propagates along the z axis to the right. The parameter (cid:96)
0
−
determines the magnitude of such a maximum. The structure of such solutions has been
extensively studied in [22,23] in conjunction with particular choices of Π together with
µν
generalizations discussed in [24,25].
In general the six anti-symmetric tensors with components δµδν in a Minkowski
[γ σ]
Cartesian coordinate system are covariantly constant and can be used to construct
a complex eigen-basis of antisymmetric chiral tensors Πs,κ, with s CE,CM and
∈ { }
κ 1,0,1 , satisfying
∈ {− }
Πs,κ = κΠs,κ (3)
z
O
where the operator represents θ rotations about the z axis generated by i∂
z θ
O − −
on tensors . These in turn can be used to construct a complex basis of chiral eigen-
‡
Maxwell tensor fields Fs,κ. The index s indicates that the CE (CM) chiral family
µν
contain electric (magnetic) fields that are orthogonal to the z axis when κ = 0.
−
The chiral eigen-fields Fs,0 inherit the axial symmetry of α(t,r,z) while those with
µν
In terms of the Lie derivative, = i and ΠCE, 1 =d(x iy) dt, ΠCE,0 =dz dt,ΠCM,κ =
‡ Oz − L∂θ ± ± ∧ ∧
(cid:63)ΠCE,κ where x=rcos(θ),y =rsin(θ)
4
(CM,κ=1) (CM,κ=0) (CM,κ= 1)
−
Figure 2: Three-dimensional spacecurves for particles subject to an incident (CM,1)
laser pulse (left), (CM,0) laser pulse (centre) and (CM, 1) laser pulse (right) with
−
parameters Λ = 600,Ψ = 1,Ψ = 1000,Φ = 0.001,Ξ = 1 . Each particle has initial
1 2
velocity R˙({0) = 0,θ˙(0) = 0,Z˙(0) = 1 . The shaded circul}ar disc region indicates the
{ 200}
initial spot size (R = 10000 for (CM, 1) laser pulses and R = 20000 for a (CM,0) laser
±
pulse) relative to the black markers on the spacecurves that denote the initial positions
of the charged test particles.
κ = 1 do not. The directions of electric and magnetic fields in any of these Maxwell
±
solutions depend on their location in the pulse and the concept of a pulse polarisation
is not strictly applicable. The chiral content as defined here can be used in its
place. Non-chiral (complex) pulse configurations can be constructed by superposition:
(cid:80) (cid:80)
F = Fs,κ s,κ with arbitrary constant complex coefficients s,κ.
µν s κ µν C C
The energy, linear and angular momentum of the pulse in vacuo can be calculated from
the components T of the Maxwell stress-energy tensor T = 1g αβ α
µν µν −4 µνF Fαβ−FµαF ν
where = Re(F ). If e and b denote time-dependent real electric and magnetic
µν µν
F
3-vector fields associated with any pulse solution F , its total electromagnetic energy
µν
, for a fixed set of parameters and any z, is calculated from
J
(cid:90) (cid:90)
1
∞
= dt (e b) dS (4)
J µ × ·
0 S
−∞
where S can be any plane with constant z = z > 0. For spatially compact pulse fields
0
in vacuo this coincides with the total pulse electromagnetic energy
(cid:90) (cid:90) (cid:90) 2π (cid:90)
∞ ∞
= ρd = dz dθ rdr ρ(t,r,z,θ) (5)
E V
0 0
V −∞
(cid:16) (cid:17)
whereρ 1 (cid:15) e e+ bb isintegratedoverallspace . Thisfollowssince (e b) =
≡ 2 0 · µ·0 V ∇· ×
µ ∂ ρ. To correlate with other laser pulse properties and the choice of parameters,
0 t
− J
we bring the pulses Fs,κ for various values of s and κ into classical interaction with
µν
one or more charged point particles. The world-line in spacetime of a single particle,
parameterized in arbitrary coordinates as xµ = ξµ(τ) with a parameter τ, is taken as a
5
solution of the coupled non-linear differential equations
q
(τ) = (ξ(τ))Vν(τ) (6)
Aµ m c2Fµν
0
in terms of the particle charge q and rest mass m , for some initial conditions ξ(0),V(0),
0
where the particle 4-velocity satisfies Vν V = 1 and its 4-acceleration is expressed in
ν
−
terms of the Christoffel symbols Γδβ as (τ) = ∂ V (τ)+V (τ)V (τ)Γδβ (ξ(τ)). In
µ µ τ µ δ β µ
A
the following, radiation reaction and inter-particle forces are assumed negligible. From
the solution ξ(τ) one can determine the increase (or decrease) in the relativistic kinetic
energy transferred from the electromagnetic pulse to any particle and the nature of its
trajectory in the laboratory frame for different choices of s and κ.
3 Interactions of Compact Laser Pulses with Matter
Theanalysisoftheprevioussectioncanbeusedtoinvestigatethebehaviourofaclassical
laser pulse interacting with electrically charged matter and motivates a model for a
quantum laser pulse interacting with electrically neutral matter.
In order to facilitate the classical behaviour, we reduce the above equations of motion to
dimensionlessformandfixthephysicaldimensionsofthefieldsinvolved. TheMinkowski
metric tensor field g = g dxµdxν (with g = diag( 1,1,1,1)) in inertial coordinates
µν µν
−
x0 = ct,x1 = x,x2 = y,x3 = z) has SI physical dimensions [L]2. The SI dimension
of electromagnetic quantities follows by assigning to (cid:15) F dx[µdxν] in any coordinate
0 µν
system the physical dimension of charge. Furthermore, in terms of Minkowski polar
coordinates t,r,z,θ , introduce (for ease of graphical visualization) the dimensionless
{ }
coordinates R = r/Φ(cid:96) , T = ct/(cid:96) , Z = z/Ξ(cid:96) and real dimensionless parameters
0 0 0
{ }
Λ,Ψ = ψ /(cid:96) (j = 1,2) where [Ψ ] = [Φ] = [Ξ] = 1,[(cid:96) ] = [L]. Then with the
j j 0 j 0
dimensionless complex scalar field α(T,R,Z) = α(t,r,z) and Greek indices ranging
(cid:98)
over T,R,Z,θ with (cid:15)T,R,Z,θ = 1 , we write
{ }
m c2(cid:96)3 (cid:16) (cid:17)
A = 0 0 RΛ∂ αΠ(cid:98) (cid:15)γµν (7)
δ q γ (cid:98) µν δ
for a choice of dimensionless covariantly constant tensor Π(cid:98) (T,R,Z,θ) = Π (t,r,z,θ),
µν µν
so that [(cid:15) A dxµ] has the physical dimension of electric charge. The total power density
0 µ
ˆ
P and total energy density are now defined by
E
(cid:90) (cid:90) (cid:90) 2π
∞ ∞
= dT dR dθP(T,R,Z,θ)
J
0 0
−∞
(cid:90)
∞ ˆ
= dZ (T,Z).
E E
−∞
6
The real parameter Λ controls the strength of the electric and magnetic fields in Fs,κ
µν
for fixed values of the real parameters Ψ ,Ψ ,Φ,Ξ and the overall scale (cid:96) will be
1 2 0
fixed in terms of the total electromagnetic energy of the pulse. For a choice of such
parameters the associated real fields e and b enable one to calculate a numerical
value Γ such that = (cid:96) Γ. The diffraction of the pulse peak along the z axis
0
J −
can be used to define a pulse range relative to the maximum of the pulse peak at
z = 0. To this end, the density (cid:98)(T,Z) defines the dimensionless range Z by
rg
E
(cid:98)(0,0)/(cid:98)(T ,Z ) = 2, where the peak at Z = Z > 0 and T = T > 0 is half the
1 rg rg 1
E E
height of the peak at Z = 0,T = 0. If during the interval [0,T ] the pulse propagates
1
with negligible deformation in Z, one may estimate its width Z at half height and
w
the dimensionless pulse axial speed β = Z /T . This yields the dimensionless pulse
rg 1
duration or temporal width T = Z /β. From these dimensionless values one deduces
0 w
the pulse SI characteristics in terms of (cid:96) and hence . If the picosecond is used
0
J
as a unit of time, the pulse duration becomes t = (cid:96) T /c = (cid:96) Z /(βc) = N 10 12
0 0 0 0 w −
s for some value N and hence (cid:96) = (cβN/Z )10 12 m, = (ΓβcN/Z )10 12 J,
0 w − w −
J
z = Ξcβn10 12 m and z = (cid:96) ΞZ = (ΞβcNZ /Z )10 12 m. A dimensionless
w − rg 0 rg rg w −
spot-size of the pulse at Z = Z > 0, T = Z /β is then determined by the behaviour
0 0
of P(R,Z /β,Z ,θ). At each value of Z this function of R and θ has a clearly defined
0 0 0
principal maximum. If one associates a circle of dimensionless radius R (Z ) with
s 0
such a maximum locus it can be used to define a spot-size at z = z with radius
0
r (z )=(cid:96) ΦR (Z )=(cβNΦR (Z )/Z )10 12 m. Figure1displaysaclearlypronounced
s 0 0 s 0 s 0 w −
principle maximum in the power density profile P as a function of X = Rcos(θ) and
Y = Rsin(θ) at Z = 0, T = 0 for a specific choice of the parameters (Λ,Ψ ,Ψ ,Φ,Ξ).
1 2
The same parameter set is used to numerically solve (6) for a collection of trajectories
for charged particles, each arranged initially around the circumference of a circle in a
plane orthogonal to the propagation axis of incident CM type laser pulses with different
chirality κ. The resulting space curves in 3-dimensions, displayed in figure 2, clearly
exhibit the different characteristic responses to CM pulses with distinct chirality values.
The instantaneous specific relativistic kinetic energy of a particle with laboratory speed
s κ (J) Λ z (m) t (ps) z (m) r (0) (m) I (Wcm 2)
rg 0 w s −
E
CE 0 8.07 40000 1.44 3 8.99 10 4 0.018 2.646 1012
−
× ×
CE 1 7.69 1 2.43 3 8.99 10 4 0.009 1.009 1012
−
± × ×
CM 0 8.41 1 0.899 2 6.00 10 4 0.012 9.304 1011
20 × − ×
CM 1 7.69 1 1.80 3 8.99 10 4 0.009 1.009 1012
−
± × ×
CM 1 76903 100 1.80 3 8.99 10 4 0.009 1.009 1016
−
± × ×
Table1: TableshowingSIlasercharacteristicsforvarious(s,κ)laserpulseconfigurations
with parameters Ψ = 1,Ψ = 1000,Φ = 0.001,Ξ = 1 .
1 2
{ }
7
104
Λ=1000 103 (CM,0)
103 Λ=100 (CE,0)
1 Λ=10 (CM, 1)
γ−102 10−1 (CE,±±1)
101
10 5
−
100 103 106 100 103 106
T T
Figure 3: Specific kinetic energy transfer γ 1 to any charged test particle. On the
−
left, the (CM,1) pulse has parameters Ψ = 1,Ψ = 1000,Φ = 0.001,Ξ = 1 . On the
1 2
{ }
right, relative values of γ 1 are displayed for various (s,κ) pulses with parameters
−
Λ = 1,Ψ = 1,Ψ = 1000,Φ = 0.001,Ξ = 1 . The differences between the
1 2
{ }
energy transfers for some pulses appear indistinguishable relative to others owing to
the logarithmic scales employed. In all cases, the charged particle has initial position
R(0) = 1,θ(0) = π,Z(0) = 1 and initial velocity R˙(0) = 0,Z˙(0) = 1 ,θ˙(0) = 0 .
{ 2 } { 200 }
(cid:113)
v is γ 1 in terms of the Lorentz factor γ given by γ 1 = 1 v2. In figure 3, this
− − − c2
quantity is displayed as a function of T on the left for a charged particle accelerated by a
fixed chirality (CM, 1) type pulse where the pulse energy is varied by changing Λ. On
−
the right the energy transfer dependence on pulse chirality for both CE and CM type
pulses with fixed laser energy is displayed. We deduce that the pulse momentum and
angular momentum [26] in the propagation direction can transfer an impulsive force
and torque respectively to charges lying in an orthogonal plane. More generally, the
classical configurations of a high energy pulse labelled CE or CM could be distinguished
experimentally by their interaction with different arrangements of charged matter.
Furthermore, by a suitable choice of parameters, (CE, κ) type modes can be constructed
that yield the same physical properties ( ,z ,z ,β) for all κ. Similarly the (CM,κ)
rg w
J
type modes yield a κ independent set with physical properties distinct from those
determined by the (CE,κ) modes. The pulse group speed magnitudes (as defined
above) of all these configurations are determined numerically and are bounded above
by the value c. To illustrate some of these statements, Table 3 summarizes the SI laser
pulse characteristics determined by solving the system of equations in (6) for a specific
choice of Ψ ,Ψ ,Φ,Ξ and various values of Λ, selected to demonstrate the physical
1 2
{ }
characteristics of existing laser pulses. We conclude that both the CE and CM type
pulses can propagate localized packets with linear and angular momentum analogous
to the quantum characteristics of propagating wave packets describing particles with
different masses.
These observations suggest that, for electromagnetic micropulses with t (cid:46) (cid:126),
0
E
a parameterized effective quantum Hamiltonian H describing a particular pair of
8
interacting massive point particles may provide the simplest effective description of
general non-stationary quantum states of a laser pulse. Such states are then defined
as elements of a complex Hilbert space = L2(R3,C3) L2(R3,C3) where each factor
denotes the space of complex square inHtegrable 3-comp⊗onent vectors on R3 carrying
irreducible finite dimensional representations of the 3-dimensional rotation group .
§
Such a construction is analogous to the Hilbert space used in Bohm and
vib rot
H ⊗ H
Loewe’s dynamic model of a vibrating and rotating diatomic molecule after removing
its translational degrees of freedom [27]. The space is spanned by eigenstates of
vib
H
the Hamiltonian H for a one-dimensional simple harmonic oscillator parameterized
vib
by a fundamental frequency ω and the space is spanned by eigenstates of the
0 rot
H
Hamiltonian H for a one-dimensional rigid rotator parameterized by a constant
rot
moment of inertia parameter . The simplest model for the diatomic molecule takes for
I
its Hamiltonian:
H = H I +I H + W W
mol vib rot vib rot
⊗ ⊗ G ⊗
in terms of the three phenomenological real parameters ω , , and where I denotes the
0
I G
identity operator on the relevant space. For certain diatomic molecules this idealized
model (and its refinements) accounts for well-established empirical formulae for their
electronic energy states. States in the space must describe the translational modes of
H
a quantum packet as well as possible mode-mixing. To this end we model their unitary
evolution generated by an effective Hermitian Hamiltonian of the form:
H = H (x ,x )+H (x ,x )
0 1 2 int 1 2
where
(cid:18) (cid:126)2 (cid:19) (cid:18) (cid:126)2 (cid:19)
H = 2 I +I 2
0 −2µ ∇x1 ⊗ ⊗ −2µ ∇x2
1 2
with real parameters µ ,µ > 0. For quantum pulses that are deemed relativistic, one
1 2
can include relativistic corrections by the replacement
(cid:126)2 (cid:113)
2 (cid:126)2c2 2 +µ2c4 µ c2
−2µ ∇xn −→ − ∇xn n − n
n
for n = 1,2 and working in a momentum representation. The structure of H (x ,x )
int 1 2
dependsuponthephysicalpropertiesofthemediumthroughwhichthepulsepropagates.
A simple model that accommodates quantum birefringence, medium anisotropy and
inhomogeneity may be constructed in terms of parameterized Hermitian operators
W (x ), W (x ):
1 1 2 2
H = W (x ) W (x ).
int 1 1 2 2
⊗
The inner product on = is defined by φ φ ,φ φ = φ ,φ φ ,φ for
§ H H1 ⊗H2 (cid:104) 1 ⊗ 2 (cid:48)1 ⊗ (cid:48)2(cid:105) (cid:104) 1 (cid:48)1(cid:105)H1(cid:104) 2 (cid:48)2(cid:105)H2
φ ,φ , n=1,2 in terms of the inner products φ ,φ on
n (cid:48)n ∈Hn (cid:104) n (cid:48)n(cid:105)Hn Hn
9
Thesixclassicalchiralstateslabelled(s,κ)thatevolveaccordingtotheclassicalMaxwell
equations are now replaced by elements Ψ satisfying the Schr¨odinger equation:
t
∈ H
(cid:126) ∂
Ψ = HΨ
−i ∂t t t
with Ψ prescribed at any time t and satisfying Ψ ,Ψ = 1. In a direct product
t0 0 (cid:104) t0 t0(cid:105)
matrix representation, decomposable states in can be written α(t,x ) β(t,x ) with
1 2
H ⊗
   
α (t,x ) β (t,x )
1 1 1 2
   
   
α(t,x ) = α (t,x ) and β(t,x ) = β (t,x ).
1  2 1  2  2 2 
   
α (t,x ) β (t,x )
3 1 3 2
By analogy with the description of non-relativistic spin 1 qubit states, call α the
−2
left qutrit component of such a bi-qutrit state and β the corresponding right qutrit
component. The dynamic description of a general state ψ requires an explicit
t
∈ H
interaction Hamiltonian operator. To model interactions with a classical medium we
envisage here a smooth fabricated meta-material with specified inhomogeneous and
anisotropic characteristics. This is similar to the specification of the magnetic moment
interactionofthequbitstates(duetoanunpairedbounds-waveelectron)ofasilveratom
with a classical (inhomogeneous) static magnetic field in the Stern-Gerlach experimental
arrangement. An important practical distinction however arises since the laser states,
unlike electrons, are electrically and magnetically neutral and hence their interaction
with electromagnetically neutral properties of a fabricated meta-material may be more
versatile than in the Stern-Gerlach situation [28].
Consider, for example, the evolution of the left qutrit component α(t,x ) under the
1
Hamiltonian:
(cid:126)2
2 I +G S N(t,x ) I
−2µ ∇x1 ⊗ 1 · 1 ⊗
1
where the constant (3 3) Hermitian matrix-valued vector S satisfies the commutation
×
relation S S = iS, the 3-vector N(t,x ) is a classical field of unit-vectors on some
1
×
fabricated meta-material and G is a parameter with the physical dimensions of energy.
1
Such a Hamiltonian assigns a single preferred field of spatial directions N(t,x ) for the
1
interaction of quantum states with the medium. More complex media could involve
multiple anisotropies described by a multi-directional set of unit-vector fields.
Given α(0,x ) with
1
3 (cid:90)
(cid:88)
α (0,x ) 2d3x = 1
j 1 1
| |
V
j=1
10