text:
# Confinement of charged tachyons in AdS spacetime

It is well known that the negative cosmological constant of AdS spacetime can act like a confining potential. That is, in contrast to asymptotically flat spacetime, in an asymptotically AdS spacetime massive particles cannot escape to infinity. However, massless particles can escape to infinity and actually do so in a finite time.

As tachyons travel faster than massless particles, is it true that all tachyons can escape to infinity as well?

If the answer is yes, then I have some trouble understanding the following argument in a paper by Horowitz on holographic superconductivity (see here). Here, the considered action of the holographic dual to the superconductor (the bulk action) is

$S=\int d^4x\sqrt{-g}\left(R+\frac{6}{L^2}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-|\nabla\Psi-iqA\Psi|^2-m^2|\Psi|^2\right)$,

i.e., a complex scalar field $\Psi$ and a Maxwell field $A_t$ (electric) coupled to gravity. The effective mass for $\Psi$ following from this action is $m_{eff}^2=m^2+q^2g^{tt}A_t^2$.

In constructing this dual theory, Horowitz argues that "In AdS, the charged particles cannot escape, since the negative cosmological constant acts like a confining box, and they settle outside the horizon." (Of course, only for particles for which the sign of the charge is the same as that of the black hole.)

However, the case considered subsequently is $m^2=-\frac{2}{L^2}$, which implies that $m_{eff}^2<0$ since also $g^{tt}<0$. Hence they consider tachyons!

Being tachyons, how can these particles settle outside the horizon? Why would they be confined by the cosmological constant rather than escape to infinity?

EDIT: right now I'm actually questioning my claim that all tachyons automatically travel faster than light...

• Could you please give a link to the paper? – Frederic Brünner Jul 7 '14 at 12:04
• @FredericBrünner I added the link. Specifically, the quote and the choice of $m^2$ are on page 6. – ScroogeMcDuck Jul 7 '14 at 16:04


text:
# Explain how (or if) a box full of photons would weigh more due to massless photons

I understand that mass-energy equivalence is often misinterpreted as saying that mass can be converted into energy and vice versa. The reality is that energy is always manifested as mass in some form, but I struggle with some cases:

Understood Nuclear Decay Example

In the case of a simple nuclear reaction, for instance, the total system mass remains the same since the mass deficit (in rest masses) is accounted for in the greater relativistic masses of the products per $E=\Delta m c^2$. When a neutron decays and you are left with a fast proton and a relativistic electron. If you could weigh those two without slowing them down, you would find it weighed as much as the original neutron.

Light in a Box

This becomes more difficult for me when moving to massless particles like photons. Photons can transmit energy from one heavy particle to another. When a photon is absorbed the relativistic mass (not the rest mass) of the (previously stationary) particle that absorbs it increases. But if my understanding is correct, the energy must still be manifested as mass somehow while the photon is in-flight, in spite of the fact that the photon does not have mass.

So let's consider a box with the interior entirely lined with perfect mirrors. I have the tare weight of the box with no photons in it. When photons are present the box has an additional quantifiable amount of energy (quantified below) due to the in-flight photons. Say there are $N$ photons... obviously assume $N$ is large.

$$\Delta m = \frac{ E }{ c^2 } = \frac{ N h }{ \lambda c}$$

Interactions are limited to reflections with the wall, which manifest as a constant pressure on the walls. If I hold this box in a constant gravitational field (like the surface of Earth) then there will be a gradient in the pressure that pushes down slightly. Is this correct? Wouldn't there still technically be mass as the photons are in-flight, which would cause its own gravitational field just as all matter does? How is this all consistent with the assertion that photons are massless? Is it really correct to say that photons don't have mass? It seems to be a big stretch.

Please offer a more complete and physically accurate account of this mirror-box.

-

The statement that photons are massless means that photons do not have rest mass. In particular, this means that, in units where $c=1$, the magnitude of the photon 3-momentum must be equal to the total energy of the photons, rather than the standard relationship where $m^{2} = E^{2}-p^{2}$.

But, you can create multi-photon systems where the net momentum is zero, since momentum adds as a vector. When you do this, however, since the energy of a non-bound state is always non-negative, the energies just add. So, this system looks just like the rest frame of a massive particle, which has energy associated with its mass and nothing else.

The statement about gravity is a little bit more subtle, but all photon states will interact with the gravitational field, thanks to the positive results of the light-bending observations that have been made over the past century. So you don't even need a construction like this to get photons "falling" in a gravitational field.

-
Would it then be correct to say that the relativistic mass, not the rest mass, of a photon is $m=\frac{h}{\lambda c}$? Then all statements I would make about the mirror box could be done using this. I think that $(m_0 c^2)^2 = E^2 - (p c)^2$ (this might be what you had in mind) still holds, however, since $m_0=0$. –  Alan Rominger May 31 '11 at 14:46
@Zassoundtsukushi: YOu can do that, but it's somewhat conceptually complicated to use the term "relativistic mass" eventually--the "relativistic mass" is really just the energy, and has properites closer to an allergy. –  Jerry Schirmer May 31 '11 at 15:00
Yes, to the extent that your unit system allows it (like $\frac{MeV}{c^2}$ for mass) I agree that the energy of the photons can be looked at as "just the energy". My question is sufficiently answered by nothing that this photon relativistic mass or energy (which one you use is semantics) exhibits all of the properties expected of that quantity of mass. This means that it is affected by gravitational fields and warps space-time itself. This is more than I previously felt comfortable claiming, but the conceptual picture here seems to be consistent. –  Alan Rominger May 31 '11 at 15:45
@Zassoundsukushi: the reason why it won't work out is technical--radiation gravitates, but not in the same way as matter, and rest mass and relativistic energies don't add together in the same way when you're creating systems out of many particles. At a first glance, you're ok using them semi-interchangeably, but realize that it's complication-prone if you plan on going deeper into this stuff. –  Jerry Schirmer May 31 '11 at 16:28
@Jerry "not in the same way as matter" -> how so? Gravitation couples to stress energy tensor which doesn't really differ all that much between massless and massive systems... –  Marek May 31 '11 at 19:04

I think that there is some confusion in your understanding of relativistic physics in the statement here:

In the case of a simple nuclear reaction, for instance, the total system mass remains the same since the mass deficit (in rest masses) is accounted for in the greater relativistic masses of the products per E=Δmc2. When a neutron decays and you are left with a fast proton and a relativistic electron. If you could weigh those two without slowing them down, you would find it weighed as much as the original neutron.

The correct statement is that the summed four vector of all the decay products would have the effective mass of a neutron.

Masses are not conserved in relativistic physics, in an analogous way that lengths are not conserved when adding vectors in three dimensions. What is conserved is energy and momentum, a four vector whose measure is m*c, where m is the effective mass of the system, similar to the length of a three-vector after the addition of three-vectors .

When a pi0 goes into two photons, it is true that the available energy for each gamma in the centre of mass system of the pio is half the mass of the pio, and the measure of the invariant mass of those two gammas will be the mass of the pio. When more particles are involved , lets say two pio's then the invariant mass of the four gammas four momenta is not additive to two pio masses. It is better to forget about convoluted arguments with masses and think of four momenta when in the relativistic regime.

Now a box of photons will have a four momentum sum in measure equal to E*2/c*2 - |p|*2=m*2*c**2. (please see the wiki link for clear terminology)

If the three vector momentum sums up to zero, the effective mass of the photons in the box will be E/c**2. Small but there to be weighed.

-
There is no confusion, these statements are unambiguously 100% correct, and are the reason physicists used relativistic mass as a concept. –  Ron Maimon Jul 4 '12 at 5:50

There are no confusions in your understanding, everything you said is correct, and it is the nontrivial content of Einstein's E=mc^2 paper. These systems are the reason that "relativistic mass", as introduced by Tolman is pedagogically useful. The concept that we call "mass" in our day-to-day life is the energy of a system (in mass units), and when you only use the word mass to mean "rest-mass", the intuitive concept is changed somewhat.

For the atomic fission, the fast moving fragments have energy which is equal to the initial bomb energy. If you weigh them without slowing them down (for example, if they are charged and you capture them by making them do circles in a magnetic field), the weight you would field on a scale once they are captured would increase by the relativistic mass (the energy over c^2).

The photons in a spherical mirror box weigh the box down exactly as the relativistic mass of the photons inside. The pull of the Earth on these photons is on the relativistic mass. If you replaced the photons with a particle gas at the same pressure, and removed a little mass from the walls to make the total mass be positive, the gravitatonal field outside will be the same, this isn't true replacing the photons with a pressureless block with a weight equal to their relativistic mass, only because the pressure contributes to the gravitational field too.

-

## protected by Qmechanic♦Mar 14 '14 at 6:51

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text:
Get direct links to references, BibTeX extraction and comments on all arXiv papers with: Librarian
Elementary Thermal Operations
Matteo Lostaglio
1
,
´
Alvaro M. Alhambra
2
, and Christopher Perry
3
1
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Castelldefels (Barcelona), 08860, Spain
2
Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
3
QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark
February 5, 2018
To what extent do thermodynamic resource
theories capture physically relevant constraints?
Inspired by quantum computation, we deﬁne
a set of elementary thermodynamic gates that
only act on 2 energy levels of a system at a
time. We show that this theory is well re-
produced by a Jaynes-Cummings interaction in
rotating wave approximation and draw a con-
nection to standard descriptions of thermalisa-
tion. We then prove that elementary thermal
operations present tighter constraints on the al-
lowed transformations than thermal operations.
Mathematically, this illustrates the failure at ﬁ-
nite temperature of fundamental theorems by
concerning stochastic maps. Physically, this im-
plies that stronger constraints than those im-
posed by single-shot quantities can be given if we
tailor a thermodynamic resource theory to the
relevant experimental scenario. We provide new
tools to do so, including necessary and suﬃcient
conditions for a given change of the population
to be possible. As an example, we describe the
resource theory of the Jaynes-Cummings model.
Finally, we initiate an investigation into how our
resource theories can be applied to Heat Bath
Algorithmic Cooling protocols.
1 Introduction
In recent years, ideas ﬁrst appearing in the ﬁeld of phys-
ical chemistry [14] have been used in conjunction with
information theory to analyse thermodynamics in the
quantum and nano-scale regimes. Often referred to as
the resource theory approach to quantum thermody-
namics, it has encompassed 2nd law-like statements in-
vestigating constraints on the allowed state transforma-
tions, e.g. [516], considerations on the 3rd law [1719]
and the construction of ﬂuctuation theorems [2023].
For a more thorough review of results within the ﬁeld
and beyond, see [24, 25] and references therein.
Many of these results are based upon the deﬁnition of
a restricted set of quantum operations known as ther-
mal operations (TO). These aim to capture all processes
that can be realised without an external source of work
or coherence and thus derive fundamental limitations
and bounds on thermodynamic processes and transfor-
mations. More speciﬁcally, they consist of all maps on
a system with Hamiltonian H
S
and in a state ρ that
can be written as
E(ρ) = Tr
X
U
ρ
e
βH
B
Tr [e
βH
B
]
U
. (1)
Here β = 1/(kT
B
) is a ﬁxed inverse temperature
of the environment, H
B
is an arbitrary bath Hamil-
tonian, U is an energy-preserving unitary satisfying
[U, H
S
+ H
B
] = 0, and typically (but not necessarily)
X = B, i.e. the partial trace is over the bath de-
grees of freedom. Note that if [U, H
S
+ H
B
] 6= 0, then
U requires work to be performed, as can be seen from
an argument involving the notion of passivity (see Ap-
pendix A).
Questions can be raised about the connection of this
formalism to relevant experimental scenarios. It is im-
portant to stress that giving the deﬁnition of thermal
operations in the “Stinespring form” [26] of Eq. (1) does
not imply that an agent needs full control of system-
bath interactions and arbitrary baths to implement
them; furthermore, there are instances in which a large
number of unitaries lead to the same transformation on
the system [6]. Nevertheless, it is not clear what realis-
tic constraints are captured by Eq. (1) [27]. In this pa-
per we provide answers to this question by considering
subsets of thermal operations which have clear experi-
mental realisations and contrasting them with the full
set.
In quantum computing, it is common to study which
operations can be implemented by sequences of ele-
mentary gates. Most famously, every n-qubit gate can
be implemented using sequences of one and two-qubit
gates [28]. From this perspective it is natural to ask
if every transition induced by thermal operations can
be implemented through sequences of two-level thermal
Accepted in Quantum 2018-01-14, click title to verify 1
arXiv:1607.00394v3 [quant-ph] 2 Feb 2018
Figure 1: Decomposing TO: is it possible to realise any transi-
tion allowed by TO by sequences of two-level processes?
operations, which we call elementary thermal operations
(ETOs), see Fig. 1.
We analyse the above question by comparing the pop-
ulation dynamics that can be induced by thermal op-
erations and sequences of ETOs. On the one hand,
we show that the dynamics induced by an ETO can
be satisfactorily captured within a very simple physi-
cal model, a Jaynes-Cumming interaction with a single
bosonic mode in rotating-wave approximation (RWA);
moreover, ETOs are connected to collision models. On
the other hand, we show that ETOs constitute a gen-
uinely new resource theory speciﬁcally, one that is
more restricted in the allowed transformations and eas-
ier to relate to physical models. The presence of a gap
between TOs and ETOs already at the classical level
may come as a surprise; in fact, it is based on the failure
at ﬁnite temperature of famous theorems by Muirhead
[29], Hardy, Littlewood and P´olya [30] and Birkhoﬀ [31]
for doubly-stochastic matrices [32].
Furthermore, we provide necessary and suﬃcient con-
ditions to compute the population dynamics that are
allowed in the theory of ETOs. We then use these re-
sults to investigate what further limitations, compared
to the thermal operation framework, arise in determin-
istic work processes. We show that the work of forma-
tion under ETOs can be inﬁnite, and we point out a gap
between ETOs and TOs in deterministic work extrac-
tion. This shows how restrictions in control translate
into the impossibility to achieve some of the traditional
results of single-shot thermodynamics. In addition, we
discuss the role of coherence within the resource the-
ory, showing that ETOs are more limited than thermal
operations also with regards to processing of coherence.
The present investigation can be understood as part
of a general eﬀort to develop tools to study thermody-
namic transformations under restricted control [18, 33
36]. We show that, while at inﬁnite temperature ther-
modynamic theories are largely universal, i.e. the spe-
ciﬁc constraints at hand do not matter in terms of the
thermodynamic laws arising, this is no longer the case at
ﬁnite temperature. This implies that every restriction
will lead to a diﬀerent resource theory; luckily, how-
ever, some of these can be solved using the same tools
we develop here for ETOs. For example, we can give
necessary and suﬃcient conditions for the population
dynamics achievable in the resource theory of 2-level
Jaynes-Cummings interactions in RWA, which is a the-
ory strictly contained in ETOs. This is important since,
for large enough systems and/or speciﬁc choices of phys-
ical parameters, we expect the two theories to depart
from each other. We also show how to solve a rather
general set of thermodynamic resource theories satis-
fying two assumptions: (a) partial level thermalisation
involving 2 levels at a time are (strictly) included and
(b) the set of extremal maps in the set of allowed op-
erations is known. Finally, we discuss an application
of this framework to an algorithmic cooling protocol,
showing that a particular ETO can be used to speed up
recently proposed protocols.
2 Elementary thermal operations
2.1 Deﬁnition and ﬁrst properties
A thermal operation will be called elementary if it
acts non-trivially only on a two-dimensional subspace
spanned by two eigenstates of the system Hamiltonian
H
S
. We will denote by E
i
= ~ω
i
the eigenvalues of H
S
.
Given a d-dimensional system in state ρ, denote by
p
i
the occupation probability of energy level E
i
, and let
p = (p
0
, ..., p
d1
). Note that every thermal operation E
induces a corresponding population dynamics G on p:
E(ρ) = σ Gp = q, (2)
where q is the vector of occupation probabilities q
i
of
σ and G
k
0
|k
= hk
0
|E(|kihk|) |k
0
i. G is simply a matrix
of transition probabilities G
k
0
|k
from energies {E
k
} to
{E
k
0
}. Since G
k
0
|k
0 and
P
k
0
G
k
0
|k
= 1 (because
E is completely positive and trace-preserving), G is a
stochastic matrix. Moreover, one can directly show
from Eq. (1) that G satisﬁes Gg = g, where g is the
thermal Gibbs distribution of H
S
at ﬁxed inverse tem-
perature β:
g =
e
βE
0
Z
S
, . . . ,
e
βE
d1
Z
S
, Z
S
=
d1
X
i=0
e
βE
i
. (3)
Such matrices G are referred to as Gibbs-stochastic. In
fact, the converse is true: every Gibbs-stochastic matrix
can be realised as the population dynamics of (the limit
of some sequence of) thermal operations [8, 37].
Elementary thermal operations give rise to 2-level
Gibbs-stochastic matrices G. These are a set of ma-
trices exhibiting a particularly simple structure:
1. Only two energy levels of the system (i, j) are in-
volved, i.e., G
k|k
= 1 for every k 6= i, j.
Accepted in Quantum 2018-01-14, click title to verify 2
2. Detailed balance holds, i.e. G
i|j
= e
β(E
i
E
j
)
G
j|i
.
Property 1 follows immediately from the deﬁnition of
ETOs. Property 2 follows since Gg = g is equivalent to
detailed balance in the case of 2 × 2 matrices.
Note that any population dynamics G induced by an
ETO is characterised by the two energy levels i and j
on which it acts and a single parameter describing the
“strength” of the interaction. This can be chosen to be
G
j|i
[0, 1] if E
i
E
j
. So,
G =
1 G
j|i
e
β~ω
ij
G
j|i
G
j|i
e
β~ω
ij
1 G
j|i
1
\(i,j)
,
where 1
\(i,j)
denotes the identity matrix on every en-
ergy level diﬀerent from i, j and ω
ij
:= ω
i
ω
j
is the
transition frequency. It is worth stressing that the set
of allowed operations we propose does not require the
ability to control the energy levels of the system Hamil-
tonian. We highlight that changing the energy levels
is diﬃcult in practice for many experimental scenarios,
but the ETOs do not require this ability.
Every 2-level Gibbs stochastic transition matrix can
be realised by means of ETOs, without the need for
arbitrary bath Hamiltonians:
Lemma 1. Every population dynamics satisfying prop-
erties 1 and 2 can be realised by an ETO involving a
single-mode bosonic bath.
See Appendix B for the simple proof. ETOs hence
have a much simpler structure than the generic pro-
cesses induced by thermal operations and can be im-
plemented using a simple bath. Due to Lemma 1, and
since from now on we will only focus on the population
dynamics, with some abuse of language we will refer to
the set of 2-level Gibbs-stochastic matrices (stochastic
matrices satisfying properties 1 and 2) as ETOs. Note
that this identiﬁcation is restricted to the allowed popu-
lation dynamics. ETOs and Gibbs-preserving maps on
two levels do not coincide as sets of quantum channels
(see Sec. 4.6).
Before we turn to the question of achieving ETOs
within simple physical models, let us introduce an ETO
of central importance, the β-swap, and describe how the
question posed in Fig. 1 ﬁts within the general theory
of stochastic processes.
2.2 β-swaps and partial level thermalisations
A direct swap of population between two energy levels
cannot be achieved by thermal operations, since the as-
sociated transition matrix is not Gibbs-stochastic (un-
less E
i
= E
j
or β = 0). Nevertheless, there is a trans-
formation that is the thermodynamic analogue of such
a swap: the β-swap. It corresponds to an ETO β
(i,j)
among levels i and j (E
i
E
j
) for which G
j|i
= 1:
β
(i,j)
=
1 e
β~ω
ij
1
e
β~ω
ij
0
1
\(i,j)
. (4)
An important fact regarding β-swaps is that if G is any
ETO acting on energy levels i and j, we have for some
λ [0, 1]
G = (1 λ)1 + λβ
(i,j)
(5)
i.e. every ETO can be written as a convex combination
of the identity matrix and a β-swap. This decomposi-
tion will be useful later.
Another decomposition of any ETO can be obtained
by combining β-swaps and partial level thermalisations
(PLTs). PLTs partially swap the state of two levels i, j
(E
i
E
j
) for the corresponding thermal probabilities:
(q
i
, q
j
) = (1 λ)(p
i
, p
j
) + λNg
(i,j)
, λ [0, 1], (6)
and p
k
= q
k
for k 6= i, j. Here N = p
i
+ p
j
,
g
(i,j)
=
1
1+e
β~ω
ij
,
e
β~ω
ij
1+e
β~ω
ij
. PLTs take the form
T
λ
=
1 λe
β~ω
ij
λ
λe
β~ω
ij
1 λ
1
\(i,j)
. (7)
for λ [0, 1/(1 +e
β~ω
ij
)]. If we take λ to be such that
λ = 1/(1 + e
β~ω
ij
), the ﬁnal state is proportional to a
thermal state on the corresponding energy levels and T
λ
is hence a full thermalisation on such levels. Given any
ETO G, one can directly check that either G is a PLT
or there exists a PLT T
λ
and a β-swap β
(i,j)
such that
G = T
λ
β
(i,j)
.
2.3 The inﬁnite temperature limit
A central question, introduced in Fig. 1, is whether ev-
ery population dynamics induced by thermal operations
can be realised by either
1. Sequences of ETOs.
2. Sequences of ETOs and convex combinations
thereof.
A fruitful perspective on these problems comes from the
fact that, in the inﬁnite temperature limit β 0 (or
within each energy subspace), thermal operations in-
duce population dynamics G given by doubly-stochastic
matrices. These are stochastic matrices G satisfying
G1 = 1. In this limit (studied in Ref. [38]) both of the
above questions are answered by two fundamental theo-
rems, one due to Birkhoﬀ [31] and the other to Muirhead
[29] and Hardy, Littlewood and olya [30].
Accepted in Quantum 2018-01-14, click title to verify 3
Speciﬁcally, from Eq. (5), one can recognise that
ETOs in the inﬁnite temperature limit are convex com-
binations of the identity and a swap, since β-swaps are
swaps when β 0. These matrices are simply gen-
eral 2-level doubly-stochastic matrices, also known as
T -transforms. They play a crucial role due to the fol-
lowing theorem [32]:
Theorem 2 (Muirhead [29] and Hardy, Littlewood
and olya [30]). Given p and q, there exists a doubly-
stochastic matrix G such that Gp = q if and only if
there exists a ﬁnite sequence of T -transforms G
(i)
such
that
q = G
(k)
···G
(1)
p.
At inﬁnite temperature, this theorem answers the
ﬁrst question above, since it shows that all transforma-
tions induced by thermal operations can also be realised
by ﬁnite sequences of ETOs.
If we also allow convex combinations, we can take
advantage of a fundamental theorem by Birkhoﬀ to
strengthen the above result:
Theorem 3 (Birkhoﬀ [31]). Every doubly-stochastic
matrix can be written as a convex combination of per-
mutations.
Since permutations are sequences of swaps, Birkhoﬀ’s
theorem implies that, if we allow convex combinations,
then not only do ETOs induce the same transformations
p q as thermal operations at inﬁnite temperature,
but they can in fact simulate every transition matrix
generated by a thermal operation, i.e. every doubly-
stochastic matrix.
1
2.4 Failure of ﬁnite temperature extensions:
ETO 6= TO
We can now see that asking if ETOs coincide with ther-
mal operations (in terms of the achievable transforma-
tions or in terms of the set of transition matrices) corre-
sponds to asking: do the two central theorems 2 and 3
in the theory of doubly stochastic matrices extend from
inﬁnite to ﬁnite temperatures? We will show that, in
both the weaker and stronger sense,
elementary thermal operations 6= thermal operations,
and hence both Theorems 2 and 3 fail at ﬁnite temper-
atures, i.e. for Gibbs-stochastic matrices. The conse-
quence is that ETOs deﬁne a genuinely new resource
theory of thermodynamics. We leave the proof of these
facts to Section 4 of this manuscript.
1
This is strictly stronger. In fact, the set of matrices that can
be realised through sequences of T -transforms is strictly smaller
than the set of all doubly-stochastic matrices, see [32, Chapter 2].
Knowing that ETOs provide a new resource theory
based on a simple set of allowed operations, two natural
questions arise:
1. Can we realise every ETO within simple physical
models?
2. What transformations can be realised by combin-
ing ETOs?
We begin by tackling the ﬁrst question and show that, in
contrast to thermal operations, ETOs are easily related
to simple physical models.
3 Physical models for elementary ther-
mal operations
While ETOs only require interactions with a single
mode bosonic bath (Lemma 1), we are interested here
in considering physically realistic models. This may re-
quire additional restrictions, e.g. on the dimension of
the bath or on the allowed system-bath interactions.
The results of Ref. [18] imply that not all ETOs can
be achieved with a ﬁnite-size bath. Here we consider
instead some natural limitations on the allowed system-
bath interactions. We begin this study by analysing if
the Jaynes-Cummings (JC) model reproduces the set of
ETOs to a satisfactory extent. For many practical pur-
poses, which is what interests us in providing a physical
model, we shall see that the answer is aﬃrmative.
3.1 Jaynes-Cummings model
One of the core descriptions of the interaction between
matter and radiation is the Jaynes-Cummings (JC)
model [39, 40]. Hence, we wish to know if it can provide
a natural model for ETOs.
For simplicity, consider a system with Hamiltonian
H
S
=
P
d1
k=0
E
k
|E
k
ihE
k
| whose relevant transition fre-
quencies are all well separated from each other and non-
degenerate, i.e.
E
j
E
i
6= E
j
0
E
i
0
if (j, i) 6= (j
0
, i
0
). (8)
Assume one is allowed to couple every transition ω
ij
of
the system to a single-mode bosonic bath with Hamilto-
nian H
B
=
P
n=0
n~ω
ij
|nihn|. The interaction is given
by a resonant JC Hamiltonian H
JC
in rotating wave
approximation,
H
JC
= g(σ
+
a + σ
a
).
Here a
, a are creation/annihilation operators on the
bath and σ
+
= |E
j
ihE
i
|, σ
= σ
+
excite/de-excite the
relevant transition.
Accepted in Quantum 2018-01-14, click title to verify 4
Without loss of generality let us denote the two
levels of the system involved by |E
0
i and |E
1
i and
set ~ω := E
1
E
0
> 0. One can compute ([41], Ap-
pendix E.2)
e
it
~
H
JC
=
X
n=1
1
X
k,k
0
=0
u
(n)
k,k
0
(s) |E
k
ihE
k
0
| |n kihn k
0
|
+ |E
0
ihE
0
| |0ih0|,
where s = gt/~ and, for each n, u
(n)
k,k
0
(s) are matrix
elements of the unitary
U
(n)
(s) =
cos(s
n) i sin(s
n)
i sin(s
n) cos(s
n)
.
Take the system and bath to be in the initial state
ρ e
βH
B
/Z
B
, where Z
B
= (1 e
β~ω
)
1
. One then
has that the dynamics of the system in the interaction
picture is described by
2
ρ 7→ Tr
B
e
it
~
H
JC
ρ
e
βH
B
Z
B
e
it
~
H
JC
.
Since [H
JC
, H
S
+ H
B
] = 0, this is an ETO. The trans-
formation induced on the vector p of occupation proba-
bilities is a stochastic process G whose transition prob-
abilities are:
G
1|0
(s) =
X
n=1
sin
2
(s
n)
e
β~ωn
Z
B
, G
0|0
= 1 G
1|0
,
G
0|1
(s) =
X
n=1
sin
2
(s
n)
e
β~ω(n1)
Z
B
, G
1|1
= 1G
0|1
. (9)
Of course, G
k|k
= 1 for every k 6= 0, 1. As described
before, G can be parametrised by G
0|1
[0, 1]. The
question is if this parameter interval can be covered
in the JC model. All values between 0 and the max-
imum achievable G
0|1
(s) (denoted by G
max
(β~ω)) can
be achieved, since G
0|1
(s) is a continuous function in s
and it takes the value 0 at s = 0.
First, let us investigate the low temperature limit.
For every s we have G
0|1
(s) (1 e
β~ω
) sin
2
(s).
Hence, in the zero temperature limit β ,
G
max
(β~ω) 1. In this limit every ETO can be re-
produced within the JC model with arbitrary accuracy.
Now consider any ﬁnite temperature. Setting
¯
β =
β~ω one can prove (see Appendix C)
G
max
(
¯
β)
(
1
16
8e
¯
β
e
2
¯
β
+ e
3
¯
β
+ 8
, for
¯
β [0,
log(4)
3
],
e
4
¯
β
e
3
¯
β
+ 1, for
¯
β
log(4)
3
.
(10)
2
To realise any given s, one should take g small enough and
t large enough to avoid the short times/strong couplings regime
where the model breaks down.
Figure 2: Realising ETOs in the simplest Jaynes-Cummings
model: to achieve every ETO one needs to be able to ob-
tain all de-exciting probabilities G
j|i
[0, 1], ω = ω
j
ω
i
>
0. The JC models realises every transition probability within
[0, G
max
(β~ω)], where G
max
(β~ω) lies somewhere within the
shaded area. Hence, far from the high temperature limit, al-
most all ETOs can be achieved within the JC model. Here the
blue line corresponds to the upper bound given in Eq. (10),
while the red line is the lower bound is obtained by truncating
the inﬁnite series in Eq. (9).
On the other hand, a rough lower bound on G
max
(
¯
β)
(good when temperatures are not too high) can be ob-
tained by truncating the sum in Eq. (9) at some ﬁnite
m
max
, and setting every other term to zero. We take
m
max
= 12. Then numerics suggest to choose s = 98.92.
This, together with the previous upper bounds, provides
a region where G
max
(
¯
β) must lie as a function of
¯
β, see
Fig. 2.
The ﬁnite temperature scaling is rather favourable:
at room temperature T
B
= 300K and frequencies
ω/(2π) larger than 10
13
Hz, and at millikelvin tem-
peratures with frequencies larger than 10
8
Hz, one has
G
max
(β~ω) > 0.989. Hence Fig. 2 shows that, even
though not every ETO can be realised within the JC
in RWA, a vast majority of them can be achieved, espe-
cially at lower temperatures. This establishes the con-
nection at the level of single operations,
ETOs Jaynes-Cummings model.
Of course, to implement a sequence of ETOs one still
requires considerable frequency control as we must be
able to separately couple to every transition frequency
of the system. This is feasible to the extent in which we
are interested in the thermodynamics of small enough
systems. In Section 6, we give some suggestions relating
to protocols exploiting this set of operations.
Accepted in Quantum 2018-01-14, click title to verify 5
3.2 Collision models
We now turn to investigating the connection between
the framework of ETOs and thermalisation models,
widely used in quantum thermodynamics [42].
Assume again for simplicity that Eq. (8) holds. Con-
sider a collision model of thermalisation of a system in
the presence of a large thermal bath [43]. If we describe
the bath as a collection of qubits with various energy
gaps, in this simple model a thermal particle from the
bath approaches, interacts with a two-level subspace of
the system and scatters away; the interaction conserves
energy and the particle is lost in the bath [44]. Each
“collision” is then an ETO. Reasoning as in Ref. [44],
one ﬁnds that the combined eﬀect of many identical
weak interactions on each two-level subsystem, initially
described by a population p(0), leads to an exponential
relaxation to the thermal distribution g (Appendix D):
p(t) = e
t/ξ
p(0) + N(1 e
t/ξ
)g, (11)
where ξ 0 and N is the normalisation of p(0). This
relates ETOs to the collision model.
Alternatively, from the master equation point of
view [45], we can consider Davies quantum dynami-
cal semigroups, which are routinely used as a simple
model of a system weakly interacting with a large ther-
mal bath [46], or in the low-density limit [47]. On 2-
level systems, a full characterisation of their action ex-
ists [48] and it coincides with Eq. (11). Hence, the colli-
sion model of ETOs reproduces Davies maps on qubits.
Comparing Eq. (11) with Eq. (6), we see that the
models described eﬀectively realise a subclass of ETO
dynamics, i.e. partial level thermalisations (deﬁned
in Sec. 2.2). These can be understood as Markovian
ETOs, deﬁned as the subset of ETOs whose induced
transition matrices G are embeddable stochastic ma-
trices, i.e. there is some generator L and t 0 such
that G = e
Lt
[49]. For G to be a valid stochas-
tic matrix, L must satisfy L
i|j
0 for i 6= j and
P
i
L
i|j
= 0 for all j [50]. Detailed balance is equiv-
alent to L
i|j
= e
β~(ω
i
ω
j
)
L
j|i
, where (i, j) are the two
levels involved. Computing e
Lt
p then gives Eq. (11).
In conclusion, Markovian ETOs can be related to two
diﬀerent physical models: collision models and Davies
maps. For a summary of the relations between ETOs
and physical models, see Fig. 3.
4 The resource theory of elementary
thermal operations
We now describe in more detail the resource theory of
ETOs and provide necessary and suﬃcient conditions
for a transformation to be possible under this set. In
Figure 3: ETOs and physical models: The set of ETOs is closely
reproduced by the Jaynes-Cummings model for a wide set of
physically relevant parameters and with growing precision as
temperatures lowers. The Markovian subset of ETOs, on the
other hand, is connected to simple thermalisation models.
particular, we will present explicit results for qutrit sys-
tems, since these can be readily visualised, and discuss
a typical thermodynamic task, work extraction. Let us
begin with
Deﬁnition 1 (Resource theory of elementary thermal
operations). The resource theory of ETOs allows all
transformations that can realised by sequential applica-
tion of ETOs and convex combinations thereof. When
such operations exist transforming p into q, we denote
this by p
ETO
q.
Allowing convex combination simply corresponds to
the ability to condition the choice of our protocol on
some random variable (like a coin toss). It is diﬃcult
to envisage an experimental setup in which this should
be forbidden. If no sequence of ETOs (no matter how
long) nor convex combinations thereof can transform p
into q, we will write p
ETO
9 q. We will use a similar nota-
tion if a transformation is possible by means of thermal
operations: p
TO
q (for any other set A, p
A
q).
4.1 Counterexample: ETOs do not have the
same power as thermal operations
As anticipated in Sec. 2.4, the resource theory of ETOs
is distinct from thermal operations, in that we are able
to show that there are p and q such that p
TO
q but
p
ETO
9 q.
3
To do this, let us recall the deﬁnition of a mono-
tone in a resource theory: it is a quantity that never
decreases (or never increases) under the allowed oper-
ations. We shall show that there is a monotone in the
resource theory of ETO that is not a monotone for ther-
mal operations, namely the size of the support of p.
3
A result that can be proven to be equivalent to the one we
present in this subsection was claimed in Ref. [51]. However,
to the best of our knowledge, its proof was never published [32,
Chapter 14B].
Accepted in Quantum 2018-01-14, click title to verify 6
The second concept we need to introduce to build
the counterexample is an important tool in the study
of thermal operations: thermo-majorisation. Given
an initial state p with corresponding energy levels
E = (E
0
, ..., E
d1
), the associated thermo-majorisation
curve is constructed as follows:
1. Reorder p and E so that p
i
e
βE
i
is in non-increasing
order. Such an ordering is called the β-order of p.
2. Plot the ordered points
n
P
k
i=0
e
βE
i
,
P
k
i=0
p
i
o
d1
k=0
together with the point (0, 0) and connect them
piecewise linearly to form a concave curve - the
thermo-majorisation curve of p.
Given two probability distributions p and q associated
with the same energy levels, we say that p thermo-
majorizes q if the thermo-majorisation curve of p is
never below that of q (examples are given in Fig. 4). Re-
calling that a Gibbs-stochastic matrix G with Gp = q
exists if and only if p thermo-majorises q [52], and the
fact that every such G can be realised through TOs, it
follows [8]
p
TO
q p thermo-majorises q. (12)
This implies that under TOs the height of the thermo-
majorisation curve at each value on the x-axis cannot
increase; furthermore, the decrease of these heights is
not only necessary but also suﬃcient for p
TO
q.
As ETOs form a subset of thermal operations, the
heights of a thermo-majorisation curve are also mono-
tones in the theory of ETOs. However, we shall now
Lemma 4. Given a probability distribution p, let
|supp (p) | = {number of i : p
i
> 0}. Then p
ETO
q im-
plies
|supp (p) | |supp (q) |, (13)
i.e. the size of the support is a monotone in the theory
of ETOs.
Proof. First note that given two probability distribu-
tions p
1
and p
2
and λ [0, 1]:
|supp (λp
1
+ (1 λ) p
2
) | max {|supp (p
1
) |, |supp (p
2
) |},
so taking convex combinations of ETOs can only in-
crease the size of the support. Hence, we can assume
q = Mp, with M a sequence of ETOs. Furthermore,
when combined with the decomposition of an ETO
given in Eq. (5), this implies that M can be written as
a convex combination of sequences of β-swaps. Then,
q will have support larger or equal to the support of
the distributions obtained by applying such sequences
Figure 4: Thermo-majorisation curves: We present the curves
for three probability distributions p = (0, 0.4, 0.6), q = (1, 0, 0)
and r = (0.15, 0.05, 0.8), each associated with the same Hamil-
tonian H satisfying e
βE
0
= 1.5, e
βE
1
= 0.7 and e
βE
2
=
0.3. Here, p
TO
q, as the thermo-majorisation curve of p is
never below that of q. However, since the thermo-majorisation
curve of r crosses those of p and q, we have r
TO
9 p, q and
p, q
TO
9 r. The states p and q form the basis of Corollary 5.
Note that the size of the support of p is 2, while for q it is 1.
to p. We thus need to consider only the eﬀect of β-
swaps β
(i,j)
. As the matrix associated with β-swaps,
given in Eq. (4), has full rank, it cannot decrease the
size of the support.
These considerations lead us to the following
Corollary 5. There exists p and q such that p
TO
q
but p
ETO
9 q.
Proof. Consider the three-level Hamiltonian with en-
ergy levels such that e
βE
1
+ e
βE
2
e
βE
0
together
with the initial state p such that p
0
= 0 and the ﬁnal
state q such that q
0
= 1. For example, p = (0, 0.4, 0.6),
q = (1, 0, 0) and e
βE
0
= 1.5, e
βE
1
= 0.7, e
βE
2
=
0.3, as presented in Fig. 4. As the support of q in
smaller than p, p
ETO
9 q. However, as shown in Fig. 4, p
thermo-majorizes q and hence p
TO
q.
To make the counterexample robust to the un-
avoidable experimental imprecisions, one would like a
stronger statement: there are transformations allowed
by thermal operations that cannot be approximated ar-
bitrarily well in the resource theory of ETOs. Speciﬁ-
cally, we want to exclude that there exists a sequence
of distributions q
with q
-close to q and such that
p
ETO
q
for every > 0. This stronger result follows
from the fact that set of q that can be achieved in the
resource theory of ETOs is a closed set; in turn, this is a
by-product of the considerations of the next section, in
Accepted in Quantum 2018-01-14, click title to verify 7
which we develop necessary and suﬃcient conditions for
a transformation to be possible in the resource theory
of ETOs.
4.2 Necessary and suﬃcient conditions
Given some initial state p, we will now focus on ﬁnding
all possible states that can be reached from it in the
theory of ETOs. In other words, we want to study
Deﬁnition 2 (ETO cone). Given an initial state p, let
C
ETO
(p) be the set of all q such that p
ETO
q.
The strategy will be to characterise the extremal
points of the above convex set. We can show that the
ETO cone can be completely explored by convex com-
binations of a ﬁnite number of β-swaps β
(i,j)
:
Theorem 6 (Extremal points of ETO cone). All ex-
tremal points q of C
ETO
(p) can be written as
q = β
(i
n
,j
n
)
···β
(i
1
,j
1
)
p, (14)
where n d!.
Proof. Eq. (5) allows us to distinguish the identity and
β-swaps as extremal ETOs. The ﬁrst step of the proof
will be to show that if q is extremal, then it can
be obtained through some sequence of β-swaps, as in
Eq. (14). By assumption, since q C
ETO
(p), we can
write q =
P
i
a
i
M
i
p, where each M
i
is a sequence of
ETOs and {a
i
} is a probability distribution. Each se-
quence of ETOs M
i
can contain both extremal and non-
extremal ETOs. Any non-extremal ETO can be decom-
posed using Eq. (5), obtaining
q =
X
x
b
x
β
x
p, (15)
where each β
x
denotes a sequence of β-swaps and {b
x
}
is some probability distribution, with b
x
6= 0 for all x.
We must then have β
x
p = q for every x, otherwise q is
not extremal. This completes the ﬁrst part of the proof.
The second part of the proof derives the bound n d!
on the length n of the required sequence of β-swaps. By
contradiction, assume that the length of the shortest
sequence of β-swaps β
min
such that β
min
p = q is r > d!.
For k = 1, ..., r, deﬁne q
k
= β
(i
k
,j
k
)
···β
(i
1
,j
1
)
p to be the
intermediate states obtained by applying the elements
of the sequence β
min
. Without loss of generality, we can
assume that q
k
6= q
k
0
for every k, k
0
, otherwise there
would be a sequence of length r
0
< r mapping p into q,
For any q
k
, deﬁne the β-ordering of q
k
as a vector of
indexes π
k
, obtained by a permutation of (0, ..., d 1).
Since there are only d! distinct β-orderings, after n > d!
β-swaps we necessarily have k
1
, k
2
with k
1
< k
2
and
1. q
k
1
6= q
k
2
and π
k
1
= π
k
2
,
2. q
k
2
can be obtained from q
k
1
through a thermal
operation.
Note that the second property holds trivially, since se-
quences of ETOs are also sequences of TOs and the lat-
ter are closed under composition. From the two proper-
ties above, applying Theorem 12 of Ref. [33], we deduce
that there exists a sequence of PLTs transforming q
k
1
into q
k
2
. As in the ﬁrst part of the proof, since PLTs
are non-extremal, we can decompose them through β-
swaps using Eq. (5). Hence, q
k
2
=
P
y
c
y
β
y
q
k
1
, where
as before β
y
denotes a sequence of β-swaps. This allows
us to substitute part of the sequence β
min
, obtaining an
expression with the following structure:
q = β
min
p = β
x
2
X
y
c
y
β
y
!
β
x
1
p,
where β
x
1
p = q
k
1
, β
x
2
q
k
2
= q and {c
y
} is some prob-
ability distribution with c
y
6= 0 for all y. Note that the
sum of the lengths of β
x
1
and β
x
2
is at most r1. Since
q is extremal we must have, for every y,
q = β
x
2
β
y
β
x
1
p.
Note however that there is a y such that β
y
is the iden-
tity, i.e. a sequence of length zero. It follows that there
is a sequence of β-swaps mapping p to q and of length at
most r1. This is in contradiction with our hypothesis.
Hence, it must be that n d!.
The above theorem gives, in particular, necessary and
suﬃcient conditions for determining whether p
ETO
q.
One can proceed as follows:
1. Construct the extremal points of C
ETO
(p) using
Theorem 6.
2. p
ETO
q if and only if q C
ETO
(p). Determining the
latter, given that the extremal points of C
ETO
(p)
are known, can be solved by a linear program.
We illustrate the above general results by comparing
the ETO cone with the TO cone
4
for d = 3, see Fig. 5.
We also note that Theorem 6 implies that C
ETO
(p) is
a simplex with a ﬁnite number of vertices. As such, it
is a closed set (as noted at the end of Section 4.1).
We complete the study of the main aspects of the re-
source theory of ETOs by addressing some thermody-
namically relevant questions within the new framework:
what is the maximum amount of work that can be ex-
tracted from a state? Conversely, how much work is
4
For completeness, we show how to construct the TO cone in
Appendix E
Accepted in Quantum 2018-01-14, click title to verify 8
(1,0,0) (0,1,0)
(0,0,1)
(1,0,0) (0,1,0)
(0,0,1)
Initial State, p
C
TO
(p)
C
ETO
(p)
Gibbs State, g
Figure 5: Comparison of the ETO and TO cones: Here we
plot C
ETO
(p) and C
TO
(p) for a 3-level system in state p and
with associated energy levels E on the 3 outcome probability
simplex. Note the region of states accessible using TO but not
by ETO. Here p = (0.7, 0.2, 0.1), βE = (0, 0.2, 0.5).
necessary to prepare a state? We will show that there
is a gap between what can be achieved with TO and
ETOs. As a by-product, this shows that there will also
be gaps between the predictions of the resource theory
of thermal operations and physical models in which only
ETOs are allowed.
4.3 Work in the theory of ETOs
Under thermal operations, if it is not possible to convert
p into q then an additional resource, work, may have to
be supplied to achieve the conversion. Similarly, if p can
be converted into q, it may be possible to extract work.
Within the resource theory approach, the deterministic
work can be regarded as the minimal amount of work
that must be supplied to guarantee a transformation or
the maximal amount of work that can be guaranteed
to be extracted in converting one state into another.
In this section, we investigate these concepts for ETOs
and highlight the diﬀerences with the theory of thermal
operations.
Deﬁnition 3 (Deterministic Work under ETOs).
Given two probability distributions p and q with Hamil-
tonian H
S
and a 2-level system (called a wit for short),
with Hamiltonian H
W
which has energy levels


text:
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text:
# Exact lattice supersymmetry

@article{Catterall2008ExactLS,
title={Exact lattice supersymmetry},
author={S. Catterall and D. Kaplan and M. {\"U}nsal},
journal={Journal of End-to-end-testing},
year={2008},
volume={483},
pages={71-130}
}
• Published 2008
• Physics
• Journal of End-to-end-testing
We provide an introduction to recent lattice formulations of supersymmetric theories which are invariant under one or more real supersymmetries at nonzero lattice spacing. These include the especially interesting case of N = 4 SYM in four dimensions. We discuss approaches based both on twisted supersymmetry and orbifold-deconstruction techniques and show their equivalence in the case of gauge theories. The presence of an exact supersymmetry reduces and in some cases eliminates the need for fine… Expand
Two dimensional super QCD on a lattice
• Physics
• 2017
We construct a lattice theory with one exact supersymmetry which consists of fields transforming in both the adjoint and fundamental representations of a U(Nc) gauge group. In addition to gluons andExpand
Supersymmetric lattices: A Brief introduction
Recently, new theoretical ideas have allowed the construction of lattice actions which are explicitly invariant under one or more supersymmetries. These theories are local and free of doublers and inExpand
Aspects of lattice N=4 supersymmetric Yang--Mills
Non-perturbative investigations of $\mathcal N = 4$ supersymmetric Yang--Mills theory formulated on a space-time lattice have advanced rapidly in recent years. Large-scale numerical calculations areExpand
Lattice formulations of supersymmetric gauge theories with matter fields
Certain classes of supersymmetric gauge theories, including the well known N=4 supersymmetric Yang-Mills theory, that takes part in the AdS/CFT correspondence, can be formulated on a EuclideanExpand
Nonperturbative aspects of supersymmetric quantum field theories in low dimensions
Simulations of supersymmetric field theories on the lattice with (spontaneously) broken supersymmetry suffer from a fermion sign problem related to the vanishing of the Witten index. We propose aExpand
Supersymmetric quantum mechanics on the lattice: I. Loop formulation
• Physics
• 2015
Abstract Simulations of supersymmetric field theories on the lattice with (spontaneously) broken supersymmetry suffer from a fermion sign problem related to the vanishing of the Witten index. WeExpand
Lattice Formulation of N = 2 ∗
We formulate N = 2∗ supersymmetric Yang-Mills theory on a Euclidean spacetime lattice using the method of topological twisting. The lattice formulation preserves one scalar supersymmetry charge atExpand
Spontaneous supersymmetry breaking in two dimensional lattice super QCD
• Physics
• 2015
A bstractWe report on a non-perturbative study of two dimensional N=22$$\mathcal{N}=\left(2,2\right)$$ super QCD. Our lattice formulation retains a single exact supersymmetry at non-zero latticeExpand
Supersymmetric quiver gauge theories on the lattice
A bstractIn this paper we detail the lattice constructions of several classes of supersymmetric quiver gauge theories in two and three Euclidean spacetime dimensions possessing exact supersymmetry atExpand
Supersymmetry Breaking in Low Dimensional Models
• Physics
• 2012
Abstract We analyse supersymmetric models that show supersymmetry breaking in one and two dimensions using lattice methods. Starting from supersymmetric quantum mechanics, we explain the fundamentalExpand

#### References

SHOWING 1-10 OF 169 REFERENCES
From Twisted Supersymmetry to Orbifold Lattices
We show how to derive the supersymmetric orbifold lattices of Cohen et al. (1, 2) and Kaplan et al. (3) by direct discretization of an appropriate twisted supersymmetric Yang-Mills theory. We examineExpand
Lattice supersymmetry, superfields and renormalization
• Physics
• 2004
We study Euclidean lattice formulations of non-gauge supersymmetric models with up to four supercharges in various dimensions. We formulate the conditions under which the interacting lattice theoryExpand
Supersymmetry on a euclidean spacetime lattice 1. A target theory with four supercharges
• Physics
• 2003
We formulate a euclidean spacetime lattice whose continuum limit is (2,2) supersymmetric Yang-Mills theory in two dimensions, a theory which possesses four supercharges and an anomalous global chiralExpand
Super Yang-Mills theories on the two-dimensional lattice with exact supersymmetry
We construct super Yang-Mills theories with = 2,4 supersymmetries on the two-dimensional square lattice keeping one or two supercharges exactly. Along the same line as the previous paper [1], theExpand
Critique of the link approach to exact lattice supersymmetry
• Physics
• 2007
Instituut-Lorentz for Theoretical Physics, Leiden University(Dated: February 1, 2008)We examine the link approach to constructing a lattice theory of N = 2 super Yang Mills theoryin two dimensions.Expand
Advances and applications of lattice supersymmetry
I review motivations for the study of supersymmetric field theories by lattice techniques. In particular, some of the more interesting potential applications are described. These are models ofExpand
Lattice sigma models with exact supersymmetry
• Physics
• 2004
We show how to construct lattice sigma models in one, two and four dimensions which exhibit an exact fermionic symmetry. These models are discretized and twisted versions of conventionalExpand
Supersymmetric models on the lattice
• Physics
• 1982
Abstract We construct the d =2 Wess-Zumino model and the d =4 N =2 supersymmetric gauge theory on the lattice, in the hamiltonian formalism. The models have the correct tree level continuum limit.Expand
Various super Yang-Mills theories with exact supersymmetry on the lattice
We continue to construct lattice super Yang-Mills theories along the line discussed in the previous papers [1,2]. In our construction of = 2, 4 theories in four dimensions, the problem of degenerateExpand
Lattice supersymmetry
A method is proposed for latticizing a class of supersymmetric gauge theories, including N=4 super Yang-Mills theory. The technique is inspired by recent work on “deconstruction”. Part of the targetExpand


text:
# In table, how to make one column has different font size for different line?

I use longtabu to create a table, below is the realted code

\begin{longtabu} to \textwidth {|>{\footnotesize}X[1 , p ] |>{\footnotesize}X[1 , p ] | >{\footnotesize}X[1 , p ]|}

I want the first line of the table (table head) to be footnotesize, and I need one column with tiny font size, but if i change footnotesize to tiny, all lines in that column become tiny, the first line should be footnotesize. How to do it?

• you can just start your heading cells with \footnotesize – David Carlisle Aug 25 '14 at 15:15
• Please always post the code for a complete, small document demonstrating your issue (Minimum Working Example). This is much more useful than mere fragments of code! – cfr Aug 25 '14 at 15:19
• @DavidCarlisle, THANKS! what a small change but big difference – postit Aug 26 '14 at 8:24

As noted in comments you can specify the default font for a column using the > syntax but override the font for heading or any other cells by starting the cell with a command such as \footnotesize.


text:
Reinforcement Learning
11 April 2021
Game Development . Game Testing . Machine Learning . Quality Assurance

# The Grid Sensor for Automated Game Testing

Defining visually agnostic observations for RL agents to test video games.

Ensuring quality in modern video games is a growing challenge faced by game companies. Classically, this challenge was met by game testers constantly playing the game or creating scripted behaviors to validate that a game looks and works as expected. However, as games grow in both size and complexity, it becomes infeasible to depend upon human testers and scripted behaviors alone to confirm quality. Using Reinforcement Learning (RL) to complement current approaches in game testing is a promising technique as RL agents can learn directly from playing the game without the need of human intervention. However, despite the exciting achievements applying reinforcement learning to AAA games, they have required an incredible amount of time and data with an equally impressive amount of engineering. Considering that AAA game development on its own is a colossal undertaking, minimizing the engineering cost of training and using RL agents for automated testing is vital to the success of the game project. To further complicate the issue, the game being tested is under active development. This means that game functionality can be added, changed, or removed independently of game visuals which themselves are subject to constant rapid and dramatic iterations. Thus, to use RL methods to address the growing problem of ensuring quality AAA games, models must be cheap to train and robust to change.

Much of RL research focuses on domains in which there is an assumed observation space. For example, early research in reinforcement learning using deep learning (deep RL) involved learninto play Atari 2600 games using the pixel rendering of the game. Research on these games led to many breakthroughs in terms of how to train RL agents with high dimensional input such as images.

This is a video of random agent playing the video game known as Breakout” which was originally implemented on the Atari 2600 game console.

Of the many lessons learned by this research direction, it became evident that the representation of observations can significantly affect the agent’s behavior. In practical terms, this means that the choice of what an agent can observe inherently affects the kind of behavior that an agent can demonstrate. For self-driving cars, this wisdom manifests in the use of expensive LIDAR and camera sensors to enable the on-board machine learning models to become more aware of their surroundings. For video games, howeveras all sensors to RL agents are virtualthe only restriction to the observations is their computational expense. This is evident in the popular Unity repository, ML-Agents in which virtual sensors collect observations for training and executing models.

Up until late September of 2020, there were only two kinds of sensors available in ML-Agents: cameras and raycasts. Like the Atari environment, the camera sensor provides a grayscale or RGB image as input to the agent whereas the raycast allows agents to observe and collect data about a game object down a line of sight. With direct access to the game object, the developer has control over what attributes of the game object, such as health, can be returned and observed by the RL agent. Although both types of sensors have been used in a variety of environments, they have limitations when training agentsA detailed description of these limitations is included in a Unity Blog article describing how EidosLabs, along with collaborators from Matsuko, developed the Grid Sensor to specifically address these concerns.

The Grid Sensor combines the generality of data extraction from raycasts with the computational efficiency of CNNs (Convolutional Neural Networks). The Grid Sensor collects data from GameObjects by querying the physics properties and then structures the data into a “height x width x channel” matrix. This matrix is analogous to an image from an orthographic camera but rather than representing red, green, and blue color values of objects, the “pixels” (or cells) of a Grid Sensor represent a vector of arbitrary data of objects relative to the sensor. Another benefit is that the grid can have a lower resolution, which can improve training times. This matrix can then be fed into a CNN (Convolutional Neural Network) and used for either data analysis or to train RL agents.

For information about the Grid Sensor implementation in Unity, see EidosLabs’s Pull Request on the ML-Agents repository.

Beyond the advantages of faster training times and smaller models, there are less obvious advantages for using the Grid Sensor for RL agents. The Grid Sensor uses tags, such as Enemy or Collectable, to filter what aspects of a game should be observed. When abstractions like Enemies and Collectables can be identified across different games, it allows for minimal engineering to port the agent to the new game and focus on hyperparameter tuning. Further, as the Grid Sensor can be attached to a Character controlled by a human, it provides a straightforward way to collect data from play sessions. This data can then be used for various purposes beyond testing such as understanding game dynamics, predicting player behavior, and learning policies via imitation all while remaining independent of game visuals.

Recently the Grid Sensor was ported to Unreal, validating the prior success of the Grid Sensor in Unity. During this rewriting of the Grid Sensor, it became evident that being able to query an object by a type rather than the tag of an object allows one to take advantage of the class hierarchies of objects while avoiding costly string comparisons.

Below is an example of an agent that was trained within Unreal to move towards a target within a simple Gym. The agent is given a Grid Sensor observation as well as a vector containing the direction of the goal, the direction the agent is facing, and the current distance to the goal relative to the starting distance. The actions are keyboard inputs fed directly to the Unreal PlayerControllerThis minimal example is a first step of many towards an automated game testing framework that is agnostic to game visuals and game specific actions.

A simple gym environment implemented in the Unreal game engine which requires an agent to navigate obstacles to move towards a desired goal. The agent moves towards the goal by selecting keyboard actions which are sent to the PlayerController.

Although the Grid Sensor addresses some of the issues faced when using Reinforcement Learning for automated game testing, it makes some assumptions about the kinds of tasks in which it can be used. Nevertheless, like every tool, it is not a solution to every problem, but it does shed light on both the freedom and the challenges that machine learning has in video game applications. At EidosLabs, we look forward to sharing our research journey as we aim to improve the experience of developers, designers, and the players.

## Author

Jaden Travnik joined Eidos-Montréal in 2018 as a Machine Learning Specialist. He obtained his MSc in Computing Science in 2018 under the supervision of Dr. Patrick Pilarski where his research activities were focused on Reinforcement Learning on prosthetic devices. Jaden sees how thoughtful applications of Machine Learning can remove obstacles for people while giving them more control over the things that matter to them. Video game development has many such obstacles, and it is exciting to see how Machine Learning can address these challenges.

Insert math as
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text:
# Quantum harmonic oscillator

Some trajectories of a harmonic oscillator according to Newton's laws of classical mechanics (A-B), and according to the Schrödinger equation of quantum mechanics (C-H). In A-B, the particle (represented as a ball attached to a spring) oscillates back and forth. In C-H, some solutions to the Schrödinger Equation are shown, where the horizontal axis is position, and the vertical axis is the real part (blue) or imaginary part (red) of the wavefunction. C,D,E,F, but not G,H, are energy eigenstates. H is a coherent state, a quantum state which approximates the classical trajectory.

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.[1][2][3]

## One-dimensional harmonic oscillator

### Hamiltonian and energy eigenstates

Wavefunction representations for the first eight bound eigenstates, n = 0 to 7. The horizontal axis shows the position x. Note: The graphs are not normalized, and the signs of some of the functions differ from those given in the text.
Corresponding probability densities.

The Hamiltonian of the particle is:

$\hat H = \frac{{\hat p}^2}{2m} + \frac{1}{2} m \omega^2 {\hat x}^2 \, ,$

where m is the particle's mass, ω is the angular frequency of the oscillator, $\hat x = x$ is the position operator, and $\hat p$ is the momentum operator, given by

$\hat p = - i \hbar {\partial \over \partial x} \, .$

The first term in the Hamiltonian represents the possible kinetic energy states of the particle, and the second term represents its respectively corresponding possible potential energy states.

One may write the time-independent Schrödinger equation,

$\hat H \left| \psi \right\rangle = E \left| \psi \right\rangle \, ,$

where E denotes a yet-to-be-determined real number that will specify a time-independent energy level, or eigenvalue, and the solution |ψ⟩ denotes that level's energy eigenstate.

One may solve the differential equation representing this eigenvalue problem in the coordinate basis, for the wave functionx|ψ⟩ = ψ(x), using a spectral method. It turns out that there is a family of solutions. In this basis, they amount to

$\psi_n(x) = \frac{1}{\sqrt{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{ - \frac{m\omega x^2}{2 \hbar}} \cdot H_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), \qquad n = 0,1,2,\ldots.$

The functions Hn are the Hermite polynomials,

$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{-x^2}\right)$ .

The corresponding energy levels are

$E_n = \hbar \omega \left(n + {1\over 2}\right)$ .

This energy spectrum is noteworthy for three reasons. First, the energies are quantized, meaning that only discrete energy values (integer-plus-half multiples of ħω) are possible; this is a general feature of quantum-mechanical systems when a particle is confined. Second, these discrete energy levels are equally spaced, unlike in the Bohr model of the atom, or the particle in a box. Third, the lowest achievable energy (the energy of the n = 0 state, called the ground state) is not equal to the minimum of the potential well, but ħω/2 above it; this is called zero-point energy. Because of the zero-point energy, the position and momentum of the oscillator in the ground state are not fixed (as they would be in a classical oscillator), but have a small range of variance, in accordance with the Heisenberg uncertainty principle. The zero-point energy also has important implications in quantum field theory and quantum gravity.

Note that the ground state probability density is concentrated at the origin. This means the particle spends most of its time at the bottom of the potential well, as we would expect for a state with little energy. As the energy increases, the probability density becomes concentrated at the classical "turning points", where the state's energy coincides with the potential energy. This is consistent with the classical harmonic oscillator, in which the particle spends most of its time (and is therefore most likely to be found) at the turning points, where it is the slowest. The correspondence principle is thus satisfied.

Probability densities |ψn(x)|2 for the bound eigenstates, beginning with the ground state (n = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the position x, and brighter colors represent higher probability densities.

The spectral method solution, though straightforward, is rather tedious. The "ladder operator" method, developed by Paul Dirac, allows us to extract the energy eigenvalues without directly solving the differential equation. Furthermore, it is readily generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators a and its adjoint a,

\begin{align} a &=\sqrt{m\omega \over 2\hbar} \left(\hat x + {i \over m \omega} \hat p \right) \\ a^{\dagger} &=\sqrt{m \omega \over 2\hbar} \left(\hat x - {i \over m \omega} \hat p \right) \end{align}

This leads to the useful representation of $\hat x$ and $\hat p$,

$\hat x = \sqrt{\frac{\hbar}{2m\omega}}(a+a^{\dagger})$
$\hat p = i\sqrt{\frac{m \omega\hbar}{2}}(a^{\dagger}-a) ~.$

The operator a is not Hermitian, since itself and its adjoint a are not equal. Yet the energy eigenstates |n⟩, when operated on by these ladder operators, give

$a^\dagger|n\rangle = \sqrt{n+1}\,|n+1\rangle$
$a|n\rangle = \sqrt{n}\,|n-1\rangle$ .

It is then evident that a, in essence, appends a single quantum of energy to the oscillator, while a removes a quantum. For this reason, they are sometimes referred to as "creation" and "annihilation" operators.

From the relations above, we can also define a number operator N, which has the following property:

$N = a^{\dagger}a$
$N\left| n \right\rangle =n\left| n \right\rangle$ .

The following commutators can be easily obtained by substituting the canonical commutation relation,

$[a,a^{\dagger}]=1,\qquad[N,a^{\dagger}]=a^{\dagger},\qquad[N,a]=-a,$

And the Hamilton operator can be expressed as

$H=\left(N+\frac{1}{2}\right)\hbar\omega,$

so the eigenstate of N is also the eigenstate of energy.

The commutation property yields

\begin{align} Na^{\dagger}|n\rangle&=\left(a^{\dagger}N+[N,a^{\dagger}]\right)|n\rangle\\&=\left(a^{\dagger}N+a^{\dagger}\right)|n\rangle\\&=(n+1)a^{\dagger}|n\rangle, \end{align}

and similarly,

$Na|n\rangle=(n-1)a|n\rangle.$

This means that a acts on |n⟩ to produce, up to a multiplicative constant, |n–1⟩, and a acts on |n⟩ to produce |n+1⟩. For this reason, a is called a "lowering operator", and a a "raising operator". The two operators together are called ladder operators. In quantum field theory, a and a are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy.

Given any energy eigenstate, we can act on it with the lowering operator, a, to produce another eigenstate with ħω less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to E = −∞. However, since

$n=\langle n|N|n\rangle=\langle n|a^{\dagger}a|n\rangle=\left(a|n\rangle\right)^{\dagger}a|n\rangle\geqslant 0,$

the smallest eigen-number is 0, and

$a \left| 0 \right\rangle = 0$.

In this case, subsequent applications of the lowering operator will just produce zero kets, instead of additional energy eigenstates. Furthermore, we have shown above that

$H \left|0\right\rangle = \frac{\hbar\omega}{2} \left|0\right\rangle$

Finally, by acting on |0⟩ with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenstates

$\left\{\left| 0 \right \rangle, \left| 1 \right \rangle, \left| 2 \right \rangle, ... , \left| n \right \rangle, ...\right\}$,

such that

$H \left|n\right\rangle = \hbar\omega \left(n +\frac{1}{2} \right) \left|n\right\rangle$ ,

which matches the energy spectrum given in the preceding section.

Arbitrary eigenstates can be expressed in terms of |0⟩,

$|n\rangle=\frac{\left(a^{\dagger}\right)^{n}}{\sqrt{n!}}|0\rangle$ .
Proof:
\begin{align} \langle n|aa^{\dagger}|n\rangle&=\langle n|\left([a,a^{\dagger}]+a^{\dagger}a\right)|n\rangle=\langle n|\left(N+1\right)|n\rangle=n+1\\\Rightarrow a^{\dagger}|n\rangle&=\sqrt{n+1}|n+1\rangle\\\Rightarrow|n\rangle&=\frac{a^{\dagger}}{\sqrt{n}}|n-1\rangle=\frac{\left(a^{\dagger}\right)^{2}}{\sqrt{n(n-1)}}|n-2\rangle=\cdots=\frac{\left(a^{\dagger}\right)^{n}}{\sqrt{n!}}|0\rangle. \end{align}

The ground state |0⟩ in the position representation is determined by a |0⟩ = 0,

\begin{align} &\left\langle x\left|a \right| 0 \right\rangle = 0~~~~~~~~~~\Longrightarrow\\ &\left(x + \frac{\hbar}{m\omega}\frac{d}{dx}\right)\left\langle x|0\right\rangle = 0~~~~~~\Longrightarrow\\ &\left\langle x|0\right\rangle = \left(\frac{m\omega}{\pi\hbar}\right)^{\frac{1}{4}}\exp\left(-\frac{m\omega}{2\hbar}x^{2}\right)=\psi_0 ~, \end{align}

and hence

$\langle x| a^{\dagger} |0\rangle =\psi_1 ~,$

and so on, as in the previous section.

### Natural length and energy scales

The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem. These can be found by nondimensionalization. The result is that, if we measure energy in units of ħω and distance in units of ħ/(), then the Hamiltonian becomes

$H = - {1\over2} {d^2 \over dx^2 } + {1 \over 2} x^2 ,$

while the energy eigenfunctions and eigenvalues become

$\psi_n(x)\equiv \left\langle x | n \right\rangle = {1 \over \sqrt{2^n n!}} \pi^{-1/4} \hbox{exp} (-x^2 / 2) H_n(x),$
$E_n = n + \tfrac{1}{2},$

where Hn(x) are the Hermite polynomials.

To avoid confusion, we will not adopt these "natural units" in this article. However, they frequently come in handy when performing calculations, by bypassing clutter. For example, the fundamental solution (Green's function) of H−i∂t, the time-dependent Schroedinger operator for this oscillator, simply boils down to the Mehler kernel,[4]

$\langle x| \exp (-itH) |y \rangle= K(x,y;t)= \frac{1}{\sqrt{2\pi i \sin t}} \exp \left(\frac{i}{2\sin t}\left ((x^2+y^2)\cos t - 2xy\right )\right )~,$

where K(x,y;0)=δ(x−y).

### Phase space solutions

In the phase space formulation of quantum mechanics, solutions to the quantum harmonic oscillator in several different representations of the quasiprobability distribution can be written in closed form. The most widely used of these is for the Wigner function, which has the solution

$F_n(u) = \frac{(-1)^n}{\pi \hbar} L_n\left(4\frac{u}{\hbar \omega}\right) e^{-2u/\hbar \omega} ~,$

where

$u=\frac{1}{2} m \omega^2 x^2 + \frac{p^2}{2m}$

and Ln are the Laguerre polynomials. This example shows how the Hermite polynomials and Laguerre polynomials are interrelated through the Wigner–Weyl transform.

## N-dimensional harmonic oscillator

The one-dimensional harmonic oscillator is readily generalizable to N dimensions, where N = 1, 2, 3, ... . In one dimension, the position of the particle was specified by a single coordinate, x. In N dimensions, this is replaced by N position coordinates, which we label x1, ..., xN. Corresponding to each position coordinate is a momentum; we label these p1, ..., pN. The canonical commutation relations between these operators are

$\begin{matrix} \left[x_i , p_j \right] &=& i\hbar\delta_{i,j} \\ \left[x_i , x_j \right] &=& 0 \\ \left[p_i , p_j \right] &=& 0 \end{matrix}$

The Hamiltonian for this system is

$H = \sum_{i=1}^N \left( {p_i^2 \over 2m} + {1\over 2} m \omega^2 x_i^2 \right)$.

As the form of this Hamiltonian makes clear, the N-dimensional harmonic oscillator is exactly analogous to N independent one-dimensional harmonic oscillators with the same mass and spring constant. In this case, the quantities x1, ..., xN would refer to the positions of each of the N particles. This is a convenient property of the $r^2$ potential, which allows the potential energy to be separated into terms depending on one coordinate each.

This observation makes the solution straightforward. For a particular set of quantum numbers {n} the energy eigenfunctions for the N-dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as:

$\langle \mathbf{x}|\psi_{\{n\}}\rangle =\prod_{i=1}^N\langle x_i|\psi_{n_i}\rangle$

In the ladder operator method, we define N sets of ladder operators,

$\begin{matrix} a_i &=& \sqrt{m\omega \over 2\hbar} \left(x_i + {i \over m \omega} p_i \right) \\ a^{\dagger}_i &=& \sqrt{m \omega \over 2\hbar} \left( x_i - {i \over m \omega} p_i \right) \end{matrix}$.

By a procedure analogous to the one-dimensional case, we can then show that each of the ai and ai operators lower and raise the energy by ℏω respectively. The Hamiltonian is

$H = \hbar \omega \, \sum_{i=1}^N \left(a_i^\dagger \,a_i + \frac{1}{2}\right).$

This Hamiltonian is invariant under the dynamic symmetry group U(N) (the unitary group in N dimensions), defined by

$U\, a_i^\dagger \,U^\dagger = \sum_{j=1}^N a_j^\dagger\,U_{ji}\quad\hbox{for all}\quad U \in U(N),$

where $U_{ji}$ is an element in the defining matrix representation of U(N).

The energy levels of the system are

$E = \hbar \omega \left[(n_1 + \cdots + n_N) + {N\over 2}\right]$.
$n_i = 0, 1, 2, \dots \quad (\hbox{the energy level in dimension } i).$

As in the one-dimensional case, the energy is quantized. The ground state energy is N times the one-dimensional energy, as we would expect using the analogy to N independent one-dimensional oscillators. There is one further difference: in the one-dimensional case, each energy level corresponds to a unique quantum state. In N-dimensions, except for the ground state, the energy levels are degenerate, meaning there are several states with the same energy.

The degeneracy can be calculated relatively easily. As an example, consider the 3-dimensional case: Define n = n1 + n2 + n3. All states with the same n will have the same energy. For a given n, we choose a particular n1. Then n2 + n3 = n − n1. There are n − n1 + 1 possible groups {n2n3}. n2 can take on the values 0 to n − n1, and for each n2 the value of n3 is fixed. The degree of degeneracy therefore is:

$g_n = \sum_{n_1=0}^n n - n_1 + 1 = \frac{(n+1)(n+2)}{2}$

Formula for general N and n [gn being the dimension of the symmetric irreducible nth power representation of the unitary group U(N)]:

$g_n = \binom{N+n-1}{n}$

The special case N = 3, given above, follows directly from this general equation. This is however, only true for distinguishable particle, or one particle in N dimensions (as dimensions are distinguishable). For the case of N bosons in a one dimension harmonic trap, the degeneracy scales as the number of ways to partition an integer n using integers less than or equal to N.

$g_n = p(N_{-},n)$

This arises due to the constraint of putting N quanta into a state ket where $\sum_{k=0}^\infty k n_k = n$ and $\sum_{k=0}^\infty n_k = N$, which are the same constraints as in integer partition.

### Example: 3D isotropic harmonic oscillator

The Schrödinger equation of a spherically-symmetric three-dimensional harmonic oscillator can be solved explicitly by separation of variables, see this article for the present case. This procedure is analogous to the separation performed in the hydrogen-like atom problem, but with the spherically symmetric potential

$V(r) = {1\over 2} \mu \omega^2 r^2,$

where μ is the mass of the problem. (Because m will be used below for the magnetic quantum number, mass is indicated by μ, instead of m, as earlier in this article.)

$\psi_{klm}(r,\theta,\phi) = N_{kl} r^{l}e^{-\nu r^2}{L_k}^{(l+{1\over 2})}(2\nu r^2) Y_{lm}(\theta,\phi)$

where

$N_{kl}=\sqrt{\sqrt{\frac{2\nu ^{3}}{\pi }}\frac{2^{k+2l+3}\;k!\;\nu ^{l}}{ (2k+2l+1)!!}}~~$ is a normalization constant; $\nu \equiv {\mu \omega \over 2 \hbar}~$;
${L_k}^{(l+{1\over 2})}(2\nu r^2)$

are generalized Laguerre polynomials; The order k of the polynomial is a non-negative integer;

$Y_{lm}(\theta,\phi)\,$ is a spherical harmonic function;
ħ is the reduced Planck constant:   $\hbar\equiv\frac{h}{2\pi}~.$

The energy eigenvalue is

$E=\hbar \omega \left(2k+l+\frac{3}{2}\right) ~.$

The energy is usually described by the single quantum number

$n\equiv 2k+l ~.$

Because k is a non-negative integer, for every even n we have ℓ = 0,2,...,n−2,n and for every odd n we have ℓ =1,3,...,n−2,n . The magnetic quantum number m is an integer satisfying -ℓ ≤ m ≤ℓ, so for every n and ℓ there are 2ℓ+1 different quantum states, labeled by m . Thus, the degeneracy at level n is

$\sum_{l=\ldots,n-2,n} (2l+1) = {(n+1)(n+2)\over 2} ~,$

where the sum starts from 0 or 1, according to whether n is even or odd. This result is in accordance with the dimension formula above, and amounts to the dimensionality of a symmetric representation of SU(3), the relevant degeneracy group.

## Harmonic oscillators lattice: phonons

We can extend the notion of harmonic oscillator to a one lattice of many particles. Consider a one-dimensional quantum mechanical harmonic chain of N identical atoms. This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions.

As in the previous section, we denote the positions of the masses by $x_1,x_2,...$, as measured from their equilibrium positions (i.e. $x_i=0$ if the particle i is at its equilibrium position.) In two or more dimensions, the $x_i$ are vector quantities. The Hamiltonian for this system is

$\mathbf{H} = \sum_{i=1}^N {p_i^2 \over 2m} + {1\over 2} m \omega^2 \sum_{\{ij\} (nn)} (x_i - x_j)^2 ~,$

where m is the mass of each atom (assuming is equal for all), and $\, x_i$ and $\, p_i$ are the position and momentum operators for the ith atom and the sum is made over the nearest neighbors (nn). However, as one expect, in a lattice could also appear waves that could behave as particles. For the treatment of waves, is custom to treat with the fourier space which uses normal modes of the wavevector as variables instead coordinates of particles. The number of normal modes are the same to the particles however the fourier space are very useful given the periodicity of the system.

We introduce, then, a set of N "normal coordinates" $\, Q_k$, defined as the discrete Fourier transforms of the xs, and N "conjugate momenta" Π defined as the Fourier transforms of the ps,

$Q_k = {1\over\sqrt{N}} \sum_{l} e^{ikal} x_l$
$\Pi_{k} = {1\over\sqrt{N}} \sum_{l} e^{-ikal} p_l ~.$

The quantity kn will turn out to be the wave number of the phonon, i.e. 2π divided by the wavelength. It takes on quantized values, because the number of atoms is finite.

This choice preserves the desired commutation relations in either real space or wave vector space

\begin{align} \left[x_l , p_m \right]&=i\hbar\delta_{l,m} \\ \left[ Q_k , \Pi_{k'} \right] &={1\over N} \sum_{l,m} e^{ikal} e^{-ik'am} [x_l , p_m ] \\ &= {i \hbar\over N} \sum_{m} e^{iam\left(k-k'\right)} = i\hbar\delta_{k,k'} \\ \left[ Q_k , Q_{k'} \right] &= \left[ \Pi_k , \Pi_{k'} \right] = 0 ~. \end{align}

From the general result

\begin{align} \sum_{l}x_l x_{l+m}&={1\over N}\sum_{kk'}Q_k Q_{k'}\sum_{l} e^{ial\left(k+k'\right)}e^{iamk'}= \sum_{k}Q_k Q_{-k}e^{iamk} \\ \sum_{l}{p_l}^2 &= \sum_{k}\Pi_k \Pi_{-k} ~, \end{align}

it is easy to show, through elementary trigonometry, that the potential energy term is

${1\over 2} m \omega^2 \sum_{j} (x_j - x_{j+1})^2= {1\over 2}\omega^2\sum_{k}Q_k Q_{-k}(2-e^{ika}-e^{-ika})= {1\over 2} \sum_{k}{\omega_k}^2Q_k Q_{-k} ~ ,$

where

$\omega_k = \sqrt{2 \omega^2 (1 - \cos(ka))} ~.$

The Hamiltonian may be written in wave vector space as

$\mathbf{H} = {1\over {2m}}\sum_k \left( { \Pi_k\Pi_{-k} } + m^2 \omega_k^2 Q_k Q_{-k} \right) ~.$

Note that the couplings between the position variables have been transformed away; if the Qs and Πs were hermitian(which they are not), the transformed Hamiltonian would describe N uncoupled harmonic oscillators.

The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose periodic boundary conditions, defining the (N+1)th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is

$k=k_n = {2n\pi \over Na} \quad \hbox{for}\ n = 0, \pm1, \pm2, ... , \pm {N \over 2}.\$

The upper bound to n comes from the minimum wavelength, which is twice the lattice spacing a, as discussed above.

The harmonic oscillator eigenvalues or energy levels for the mode ωk are

$E_n = \left({1\over2}+n\right)\hbar\omega_k \quad\quad\quad n=0,1,2,3 ......$

If we ignore the zero-point energy then the levels are evenly spaced at

ħω, 2ħω, 3ħω, ...

So an exact amount of energy ħω, must be supplied to the harmonic oscillator lattice to push it to the next energy level. In comparison to the photon case when the electromagnetic field is quantised, the quantum of vibrational energy is called a phonon.

All quantum systems show wave-like and particle-like properties. The particle-like properties of the phonon are best understood using the methods of second quantization and operator techniques described later.[5]

## Applications

• The vibrations of a diatomic molecule are an example of a two-body version of the quantum harmonic oscillator. In this case, the angular frequency is given by
$\omega = \sqrt{\frac{k}{\mu}}$

where μ = m1m2/(m1+m2) is the reduced mass and is determined by the masses m1, m2 of the two atoms.[6]

• The Hooke's atom is a simple model of the helium atom using the quantum harmonic oscillator
• Modelling phonons, as discussed above
• A charge, q, with mass, m, in a uniform magnetic field, B, is an example of a one-dimensional quantum harmonic oscillator: the Landau quantization.