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Open quantum systems
Marko Ljubotina
Mentor: prof. dr. Tomaž Prosen
Co-mentor: asist. dr. Ugo Marzolino
Ljubljana, 1.3.2016
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Overview
• Introduction
• Mixed states
• The density matrix
• Closed quantum systems
• Von Neumann equation
• Open quantum systems
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Dynamical maps
Quantum dynamical semigroups
Lindblad equation
Jump operators
Nonequilibrium steady state
Example
• Conclusion
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Introduction
• Closed quantum systems are an idealization
• Systems interact with the environment
• We deal with statistical ensembles
• The Markov approximation
• Extensive research in the 1970s
• Recently regained popularity with the advances in quantum
information theory
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Mixed states
• When dealing with statistical quantum mechanics or dissipative
dynamics the wave function is not sufficient
• A wave function |ψi always represents a pure state of the system
• An example of a pure state of a spin 1/2 particle:
1
|ψi = √ (|↑i + |↓i)
2
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Example - Polarization of a photon
• A single photon can have two helicities: right circular polarized |Ri or
left circular polarized |Li (or a superposition of the two)
• State of a (vertically polarized) photon exiting a laser
1
|ψi = √ (|Ri + |Li)
2
• A photon from a conventional lightbulb: 50% chance of being in the
state |Ri and 50% chance of being in the state |Li
• Measuring the probability that the photon is in the state |Ri
repeatedly would give us a 50% probability in both cases
• We can block the first photon using a polarizer but we can not block
the other photon with any polarizer with 100% efficiency
• The first is in a superposition of states, while the second can be
described by some statistical ensemble, but not with a wave function
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Mixed states
Mixed states arise when
• Dealing with systems in thermodynamic or chemical equilibrium
• Observing only a part of a larger entangled system
• Considering systems with dissipative dynamics
Any mixed state can be represented as a pure state of a larger system.
• The process is called purification
• Second photon - a simultaneous emission of two photons
1
|ψi = √ (|R, Li + |L, Ri)
2
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The density matrix
• Describes pure and mixed states with the same formalism
• The density operator of a pure state:
ρ̂ = |ψihψ|
• Generalising to mixed states:
ρ̂ =
X
pi |ψi ihψi |
i
• pi are the probabilites that our system is in the state |ψi i
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Open quantum systems
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The density matrix - properties
• Diagonal elements correspond to probabilites
• Offdiagonal elements correspond to correlations
• A trace of 1
Tr(ρ) = 1
• Self-adjoint
ρ = ρ†
• Positive-semidefinite - all it’s eigenvalues are greater or equal to 0
λi ≥ 0
• Expectation value of observables
hAi = Tr(ρA)
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Example - Polarization of a photon
Density matrices:
• In the first case
ρ = |ψihψ| =
with |ψi =
√1 (|Ri
2
1 1 1
2 1 1
+ |Li)
• And in the second
1
1
1 1 0
ρ = |RihR| + |LihL| =
2
2
2 0 1
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Closed quantum systems
• The Schrödinger equation for the dynamics of wave functions
∂
ψ(r, t) = −iHψ(r, t)
∂t
• The von Neumann equation for the Hamiltonian dynamics of density
matrices
∂
ρ = −i[H, ρ]
∂t
• The solution of the von Neumann equation is usually in the form
ρ(t0 + t) = e −iHt ρ(t0 )e iHt
• Alternatively, we define a superoperator
Hρ = [H, ρ]
which gives the following solution
ρ(t) = e −iHt ρ(0)
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Closed quantum systems
• The Hamiltonian H is self-adjoint → it’s eigenvalues are all real
• Hence the eigenvalues of
U(t) = e −iHt
are of the form
λi = e iϕi
→
||λi || = 1
• States are preserved with time - the same state will be reached again
after a sufficient amount of time has passed
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Open quantum systems
• Account for the interaction with the environment
• A different equation is needed to describe these systems
A classical example: a metallic bar connected two heat reservoirs at
different temperatures.
• The steady state is the same regardless of the initial state of the bar
• An exchange of energy between our system and the environment
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Example - spin chain
Consider a chain of N spin 1/2 particles interacting with eachother by the
Heisenberg interaction
H = −J
N−1
X
~σi · ~σi+1
i=1
We can couple this to two more spins in states of themal equilibrium
ρT =
e −βH
Tr(e −βH )
using the same interaction
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Example - spin chain
Issues
• The state of the spins in thermal equilibirum should not change if
they are to represent heat reservoirs at a constant temperature
• How does one define a temperature of a single spin
Alternative
• Creating a state with a spin flow instead
• Each spin is in a linear combination of states |↑i and |↓i
• A spin sink on the left and a spin source on the right
But we still don’t know how to approach this problem
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Dynamical maps
• Open quantum system S
• Environment system R
• Propagator
U = e −iHtot t
• We define a dynamical map Λ as
ρ → Λρ = TrR (U(ρS ⊗ ρR )U † )
Properties
• Trace preserving
• Completely positive
Λ†n = Λ† ⊗ 1n
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Quantum dynamical semigroups
A family of dynamical maps describing a system forms a semigroup:
• Λt Λs = Λt+s - the so-called semigroup condition or Markov property
• Tr((Λt ρ)A) is a continuous function of t for any density matrix ρ and
observable A
• The generator of any quantum dynamical semigroup:
∂
ρ = Lρ
∂t
•
Λt = e Lt
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Lindblad equation
Using this we derive the Lindblad equation
X
d
1 †
1 †
†
ρ = Lρ = −i[H, ρ] +
Li ρLi − Li Li ρ − ρLi Li
dt
2
2
i
• H is the hamiltonian of the open quantum system S
• L are the jump operators describing the permitted incoherent changes
in the system
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Jump operators
• A spin sink and source
+
L1 = λ1 σN
L2 = λ2 σ1−
• A totally asymmetric simple exclusion process
−
Li = λσi+ σi+1
• Dephasing noise
Li = λσiz
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Nonequilibrium steady state
• Describing the complete dynamics of a system can be challenging
• Steady states represent the state of the system after a long time
LρNESS = 0
• Depending on the system they may either be unique or form a smaller
subspace within the phase-space
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Example - spin chain
• N spins interacting through the Heisenberg interaction
H = −J
N−1
X
~σi · ~σi+1
i=1
• A spin sink and source
+
L1 = λ1 σN
L2 = λ2 σ1−
• Expand the operator L over the basis of all 2N × 2N matrices
• Solve for the eigensystem of the resulting 22N × 22N matrix and find
the eigenstates with λi = 0
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Example - spin chain
The density matrix corresponding to the unique steady state of the system with
N = 4, J = 1 and λ1 = λ2 = 0.5.
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Example - spin chain
Expected values of individual spins
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Conclusion
• Density matrices are needed in statistical quantum mechanics, open
quantum systems and other areas
• The von Neumann equation was presented
• The Lindblad equation was shown and some of the underlying
mathematical theory was outlined
The field of open quantum systems is particularly important in quantum
information theory, quantum optics and condensed matter physics.
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Thank you!
Questions?
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