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Lecture 12
Experimental realization of quantum computing: atoms
Introduction to atoms and lasers
Back to the real world:
What do we need to build a quantum computer?
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Qubits which retain their properties.
Scalable array of qubits.
Initialization: ability to prepare one certain state
repeatedly on demand. Need continuous supply of 0 .
Universal set of quantum gates. A system in which
qubits can be made to evolve as desired.
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Long relevant decoherence times.
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Ability to efficiently read out the result.
• Which atoms will make good qubits?
• How to initialize the qubits - how to cool and trap atoms?
• How to manipulate atomic states (i.e. qubits)?
• How to perform two-qubits gates - which physical processes can we use?
• How to read out the result?
• What are the error sources?
• How large are the errors?
• How can we minimize errors?
Brief introduction to a hydrogen atom
Heavy proton (put at the origin), charge e and much
lighter electron, charge -e.
Potential energy, from Coulomb's law
Potential is spherically symmetric. Therefore, solutions must have form
n is principal quantum number
spherical harmonics
l is orbital angular momentum quantum number
m is magnetic quantum number
Energy levels of the hydrogen atom
Note: energy depends only on principal
quantum number n in non-relativistic approximation.
Ground state of hydrogen atom
Ground state wave function
Hydrogen wave functions are designated by set of quantum numbers (n,l,m).
https://www.asu.edu/courses/phs208/patternsbb/PiN/rdg/electrons/fige4.shtml
https://en.wikipedia.org/wiki/Quantum_mechanics
n is principal quantum number
l is orbital angular momentum quantum number
m is magnetic quantum number
Note on spectroscopic notations (they are actually used).
There are letters associated with values of orbital angular momentum. The first few
are:
For example, state with
n=1 l=0 is referred to as 1s,
n=2 l=0 is referred to as 2s,
n=2 l=1 is referred to as 2p, and so on.
l=0 then m=0
l=1 (p state) m=-1,0,1
l=2 (d state) m=-2,-1,0,1,2
Electron has spin s=1/2, extra quantum number ms that can be 1/2 or -1/2
 s  ms  s
11
sms   
22
sms 
1
1
 
2
2
Therefor, there are two possible states for every n, l, m (i.e. for every circle in the diagram above).
Hyperfine structure
All levels of hydrogen, including
fine structure components, are
split into two more components.
So far, we ignored the spin of the hydrogen nucleus (i.e. proton), which is I=1/2. Additional
splitting of the atomic energy levels appear because of the interaction of the nuclear
moments with the electromagnetic fields of the electrons. The level splitting caused by this
interaction is even smaller than the fine structure, so it is called hyperfine structure.
Hyperfine states that are split from the ground state make particularly good qubits for
quantum information due to their long lifetimes. The hyperfine splitting of the ground state of
Cs is used to define a second.
Addition of angular momenta
Ground state of hydrogen: it has one proton with spin I=1/2 and one electron with spin J=1/2
(orbital angular momentum is zero). What is the total angular momentum F of the
hydrogen atom?
  
F  I  J
Total spin
Proton's spin, acts only
on proton's spin states
Electron's spin, acts only
on electron's spin states
The z components just add together and quantum number m for the composite
system is simply
If you combine any angular momentum I and J you get every value of angular momentum F
according to the rule:
I  J  F  I  J
I  1/ 2and J  1/ 2thenF  0,1
Note: it does not matter if it is orbital angular momentum or spin, this rule works for any angular
momentum.
Final summary: H energy level structure
Fine-structure
splitting
4.5×10-5eV
2s hyperfine splitting
7.3×10-7eV (177.6 MHz)
2s1/2-2p1/2
splitting due to Lamb shift
4.4×10-6eV
-1.8×10-4eV
3.6×10-5eV
1s hyperfine splitting
5.8×10-6eV (1420.5 MHz)
Next atom: helium
Neutral Helium (Z=2), two electrons
Hamiltonian
Two hydrogenic Hamiltonians (with Z=2), one for electron 1 and
one for electron 2.
Term that describes repulsion of two electrons.
The two electrons of the He atom are identical particles. Let's dicuss how to treat this.
Identical particles: bosons and fermions
If the particle one is in state
and particle two is in state
can be written as the simple product (we will ignore spin for now):
, then the total state
Equation (1) above assumes that we can tell which particle is particle one and which particle is particle
two. In classical mechanics, you can always identify which particle is which. In quantum mechanics, you
simply can't say which electron is which as you can not put any labels on them
to tell them apart.
There are two possible ways to deal with indistinguishable particles, i.e. to construct
two-particle wave function that is non committal to which particle is in which state:
Therefore, quantum mechanics allows for two kinds of identical particles: bosons (for the "+"
sign) and fermions (for the "-" sign). In our non-relativistic quantum mechanics we accept
the following statement as an axiom:
All particles with integer spin are bosons, all particles with half integer spin are fermions.
From the above, two identical fermions can not occupy the same state:
It is called Pauli exclusion principle.
Electrons have spin
antisymmetric.
and; therefore, are fermions. Total wave function has to be
THEREFORE, ALL ELECTRONS IN AN ATOM HAVE TO BE IN DIFFERENT
STATES - DIFFER BY AT LEAST ONE QUANTUM NUMBER FROM THE SET
n, l, m, ms
How do we label the atomic states in a general case?
The atomic state is described by its electronic configuration (1s2, for example) and a "term" symbol
that describes total S, L, and J (total angular momentum) of all electrons. The term symbol is always
written as follows:

 
J  L  S
Numbers are used for S and J but letters S, P, D, F, etc.
are used for L.
Helium energy level diagram
Helium ground state 1s2
1S
0
Helium excited states 1snl 1LJ or 3LJ
Building-up principle of the electron shell for larger atoms. Electronic
configurations and ground state terms. Hund's rules.
Pauli principle does not allow for two atomic electrons with the same quantum numbers,
therefore each next electron will have to have at least one quantum number [n, ℓ, mℓ, ms ]
different from all the other ones. We will use ↑ for ms=1/2 and ↓ for ms=-1/2.
List of distinct sets of quantum number combinations in order of increasing n and ℓ:
n
ℓ
mℓ
ms
1s
1
0
0
↑
1s
1
0
0
↓
2s
2
0
0
↑
2s
2
0
0
↓
2p
2
1
-1
↑
2p
2
1
0
↑
2p
2
1
1
↑
2p
2
1
-1
↓
2p
2
1
0
↓
2p
2
1
1
↓
1s shell (2 electrons)
2s subshell (2 electrons)
2p subshell (6 electrons)
Maximum number of
electrons in a subshell
Shells with the same n but different l (2s, 2p) may be referred to as either shells or subshells.
There are 2 electrons in n=1 shell and 8 electrons in n=2 shell.
Question for the class: what is the total number of electrons in n=3 and n=4 shells?
n=3: 3s [2], 3p[6], 3d[10], so 18
n=4: 4s [2], 4p[6], 4d[10], 4f[14], so 18+14=32
One can also arrive to this as follows:
How do these shells get filled in a periodic table, i.e. what are electronic
configurations and terms for ground states of all elements in the periodic table?
Rule 1. The Pauli principle is obeyed.
Rule 2. The total energy of all electrons is minimum for atomic ground state.
Periodic table: filling of shells
Class exercise: what are the ground-state electronic configurations for Cs (Z=55) and Tl (Z=81)?