3-1

3. Density matrices3.1 General properties

An observable's expectation value as a function of time is given by

< Ψ(t)*ΨS

| AOS |Ψ(t)

ΨS

>= < Ψ(0)*ΨH

| +iHte A −

iHte

OH

|Ψ(0)ΨH

> 3-1

where −ihHte is the time-evolution operator. As shown by the right hand side of eq. 3-1. instead

of taking the wave functions as time-dependent we can take the operator as time-dependent:

AH(t) = e+ iHtA(0)e

− iHt

3-2Taking the derivative of eq. 3-2, one obtains the Heisenberg equation of motion (compare withthe Liouville equation in section 1):

dAdt

=1i[A,H] + ∂A

∂ t 3-3

The second term is usually zero since Schrödinger operators do not usually depend explicitly ontime.

We now ask: is there an operator ˆ ρ , which is the analog of the classical distributionfunction? It should somehow be related to *Ψ Ψ , which provides the quantum mechanicalprobability distribution.

Let a system be in a pure state |Ψ〉 = ai i〉i∑ such that it can be described by a single wave

function Ψ. It then follows for the expectation value of an operator A< A >=<Ψ |A|Ψ >= aj

*aij∑

i∑ < j|A|i >. 3-4

If we define an operatorˆ ρ S =|Ψ > < Ψ| = |i > aiaj

* < j|j∑

i∑ 3-5

with matrix elements ρij = aiaj* , then the trace of the product ˆ ρ S and A is also the expectation

value of A:Tr{ˆ ρ A} = < i|Ψ > <

i∑ Ψ |A|i > = <

i∑ Ψ |A|i > < i|Ψ >=< Ψ |A|Ψ >=< A > 3-6

The density operator corresponding to state Ψ is thus the projection operator onto the presentstate of the system. Using it, we can eliminate the wavefunction from the picture and perform allnecessary quantum mechanical operations with operators alone. At some later time,

ρS (t) =|Ψ(t) >< Ψ(t)|. 3-7yielding <A(t)> if the trace is taken. Like the wavefunction, the density operator in theSchrödinger representation is explicitly time-dependent. This is unlike most Schrödingeroperators such as x or p which are not explicitly time dependent. We can also write ˆ ρ S as

ρS (t) = e− iHt |Ψ(0) >< Ψ(0)|e+ i

Ht = e−iHtρ(0)e+ i

Ht 3-8Taking the derivative of eq. 3-8, we obtain an equation of motion for the density operator

∂ρs

∂t=∂∂ t|Ψ(t) >< Ψ(t)|= ∂

∂te−iHtρ(0)e

+ iHt

∂ρS∂t

= −1i

ρS ,H[ ] , 3-9the von Neumann or quantum-Liouville equation. This looks deceptively like the Heisenbergequation of motion, but it is not. The Heisenberg equation propagates operators in the

3-2

Heisenberg representation, but ρS is in the Schrödinger representation, since it is directlyconstructed from the time-dependent wavefunction. ρ is a unique operator in that it is the"wavefunction operator". In fact, its Heisenberg operator version, like all Heisenberg operatorsdefined as ρH = exp[ +i Ht]ρS exp[

−i Ht] = ρ(0) , is completely time-independent. While ordinary

Schrödinger operators are time-independent and ordinary Heisenberg operators are time-dependent, the opposite is true for the density operator. The von Neumann equation has theopposite sign of the Heisenberg equation. In that respect, it is analogous to the time dependenceof the classical density function when compared to the Poisson-bracket equation for dynamicalvariables.

ρ also obviously satisfies the following equations:ρ† = Ψ〉〈Ψ( )† = ρ (and ρ is Hermitean) 3-10

Tr{ρ} = iΨ〉〈Ψ ii∑ = Ψ i〉〈i Ψ

i∑ = ΨΨ =1 (unit trace). 3-11

3.2 Incoherence by superposition

So far, ρ is just an alternative formulation of Ψ which allows computation of expectationvalues by trace operation. Is it more generally useful than that? The answer is yes, when onehas to deal with incoherent or "impure" states: ρ can represent the full quantum coherence of asystem. Wavefunctions are completely coherent and can represent only "pure" states. Inpractice, we have-ai ai

* diagonal elements of ρ: population in basis state i;-ai aj

* off-diag. elementsof ρ: coherence between basis states i and j.Let us see how incoherence might arise, and what it means in a simple example. Consider a

two-state system with basis states |1> and |2>, such that any state of the system can berepresentated by a wavefunctionΨ〉 = a1 1〉 + a2 2〉 . The density operator corresponding to Ψwould then be |Ψ ><Ψ |= a1a1

*|1>< 1|+a1a2*|1 >< 2|+a2a1

*| 2 ><1|+a2a2*|2 >< 2| . This can also be

verified in matrix representation by comparing the usual formula for the expectation value of anoperator A with the trace method:

A =2a1 1A1 +

2a2 2 A2 + a2a1* 1A2 + a1a2

* 2A1 3-12

Tr{ρ,A}= Trρ 11 ρ 12

ρ 21 ρ 22

⎛ ⎝ ⎜ ⎞

⎠ ⎧ ⎨ ⎩

A11 A12

A2 1 A2 2

⎛ ⎝ ⎜ ⎞

⎠ ⎫ ⎬ ⎭

= ρ11A11 + ρ12A 2 1+ ρ21A 1 2+ ρ 22A 2 2 3-13

It follows that for the two to be equal,ρ 11 = a1a1*, ρ 1 2 = a1a2

*, etc. The coefficients depend on timeif Ψ is a time-dependent wavefunction, e.g. a1 = exp[ −i

E1t] a1(t = 0)). Thus, a density matrixrepresenting a coherent quantum state Ψ〉 must have the appropriate off-diagonal matrixelements; the generalization for ∞ many basis states is clear.

Now consider a different case: Let there be N noninteracting molecules, that have beenprepared nominally in the same state, but with different phases. For instance, a short rf or laserpulse might excite dilute gas molecules in a finite sample volume at random positions, and due tothe finite speed of light, each is excited at a different time and with different relative phase by theelectromagnetic radiation. Even worse, the pulse might excite different eigenstates with differentphases if its relative phase at the two frequencies is unstable. The wavefunction for an individualmolecule numbered "n" then becomes

|Ψn >= eiϕ1na1|1 > +e

iϕ2n a2|2 > . 3-14This has the same amplitudes as before, but different random phases for the two basis states.The corresponding density matrix is

3-3

ρn = a1a1* a1a2

*ei ( 1nϕ −2nϕ )

a2a1*e−i (

1nϕ −

2nϕ ) a2a2*

⎛

⎝ ⎜

⎞

⎠ ⎟ = a1a1

* a1a2* i

1 2

(n )ϕea2a1*

− i1 2

(n )ϕe a2a2*⎛

⎝ ⎜

⎞

⎠ ⎟ 3-15

where n=1...N, the number of molecules. If we could make a measurement of the expectationvalue of A on an individual molecule "n", the only difference we would notice is that the time-dependent oscillations in <A> (eq. 3-12) have a different phase. We recover the maximumpossible information about the system. This is why "single molecule" experiments are sodesirable. In practice, one often has to measure <A> for many molecules at once, with randomlyprepared phases as in the example above, to obtain good signal. The value of A averaged overall N molecules is

<A>= 1N( Tr{ρn

n∑ A}) = Tr{1

NρnA} =

n∑ Tr(ρ A) 3-16

If the ϕi are random, then we have

→ρ =1N

ρnn∑ =

a1a1* 00 a2a2*

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 3-17

The off-diagonal elements are gone due to random phase averaging: a completely incoherentdensity matrix results. In this case, the incoherence arises not because the molecules are not allin principle individually coherent, but because our measurement process averaged over manymolecules whose relative phase was random. However, note that ρ has lost none of theproperties [1]-[4] described above:

A = Tr{ρ A}; 3-18

∂ρ ∂ t

= −1i

ρ , H[ ] ; 3-19

ρ =1N nρ

n∑ is still Hermitean; 3-20

Tr{ρ } = 1N Tr(ρn∑ ) = N

N = 1 . 3-21

We can now write a more general definition of the density operator that goes beyondwavefunctions:

Def: A Hermitean operator ρ with Tr(ρ )=1 satisfying the von Neumann eq. is a densityoperator describing some fully coherent, partially coherent, or incoherent ensemble.

There is an easy way to distinguish fully coherent ρ, which are just another representation ofsome wavefunction, from partially or completely incoherent ρ, which cannot be represented bywavefunctions and correspond to partial or complete averaging of phases. In general,

Tr{ρ2} = { ρijj∑ ρ ji

i∑ } = ρii

2

i∑ + ρij

i≠ j∑ ρij

*

= ρii2

i∑ + ( 1

N ei

ij

(n)ϕ∑

i≠ j∑ )aiaj* ( 1

N e−i

ij

(n )ϕ∑ )ajai*

3-22

Tr{ρ}2 = ( iiρ )i∑ 2 = ρii

2

i∑ + iiρ

i∉j∑ jjρ = ρii

2

i∑ + aiai

*

i≠ j∑ ajaj

* 3-23

⇒ Tr(ρ 2 ) ≤ Tr(ρ)2 =1 . 3-24

3-4

The equal sign holds only for pure states (all ij

(n )ϕ identical). Incoherence or partial coherence attime t>0 which was not present at time t<0 is always a result of an imperfect preparation ormeasurement process. Imperfect here means that we lack full knowledge of how the preparationor measurement was done. If the molecules had been strongly interacting, a fully coherent statewould of course have to be a multi-particle wavefunction with a total of 2N levels rather than just2 levels for each molecule. In that case, full coherence is even harder to measure and would beuncoverable only in a multi-quantum experiment which detected the phases of all moleculessimultaneously. In practice this is not (yet) possible, and such a system can be consideredincoherent for all practical purposes. However, there may well exist an overall systemwavefunction which is in a pure state, even though a diagonal density operator is a convenientapproximation.

The strongest type of averaging which usually occurs in natural systems is thermal averaging:Let |i> be eigenfunctions of H: H|i>=Ei|i>. If the overall state is in thermal equilibrium,

ρii = e−βE i / Q and ρij = 0 (β = 1kT

)

⇒ ˆ ρ = e− β ˆ H / Q where Q = e− βEi

i∑ . 3-25

The expectation value of A in such a state is then < A>= Tr{e−βH A}. A thermal ensemble is bydefinition completely incoherent and has zero off-diagonal density matrix elements. A given realsystem which was initially in a purely coherent state may evolve into an ensemble that is wellapproximated by eq 3-25, even though it is in principle (but not practically measurably) stillcoherent.

3.3 Incoherence by reduction

In section 3.2 we saw how a superposition of pure state density matrices with random phasesleads to a partially coherent or incoherent density matrix. In practice, such a matrix evolvesbecause a subsystem is interacting with another subsystem, and we only have access to theobservables of the first subsystem. In practice, we often call the first subsystem "the system" andthe second subsystem "the environment" or "the bath."

Clearly, this distinction is a somewhat artificial choice of the observer, based on theobservables to which the observer has access. Usually, only partial knowledge of the totalsystem is available, and little is known about the bath. Let the two subsystems be described bystates {|i>} and {|j>} in the absence of coupling. {|i>} could be the states of a molecule underinvestigation, while {|j>} are the states of a solvent bath. If the molecule and solvent areinteracting, a general wavefunction can be written Ψ = cij∑ |i > | j > . In practice, we may have noinformation about the solvent other than that it is well approximated by a thermal bath. In thatcase, we can eliminate the solvent from the density matrix by reducing the total density matrix(which may be a pure state for all we know) as follows:

ρred = Trj{ρ(i, j)}

= Trj{i,i'∑

j, j'∑ cij ci' j'

* | i' >| j' >< i |< j |}

= | i' >< i |i,i'∑ cij ci' j

*

j∑

= ci'i(red) | i' >< i |

i,i'∑ .

3-26

ρred depends only on system states |i> and their associated coordinates or momenta. The |j>degrees of freedom of the bath have been averaged out by the trace operation. One can easily

3-5

show that the new reduced density matrix still satisfies eqs. 3-20 and 3-21 by construction, butthere is a price to pay: the equation of motion of the reduced density matrix is no longer theLiouville-von Neumann equation, but rather contains new terms due to the bath which has beenintegrated out. These terms take the form of Lindblatt operators, which we will not considerfurther here. The process by which the off-diagonal elements of a reduced density matrix decayto zero under the influence of an environment is referred to as decoherence. We discuss anexample first, and then briefly sketch the derivation for the reduced equation of motion in thelimit where perturbation theory works.

3.4 Reduced density matrix for a two-level system

We will discuss reduction and decoherence using a very simple example: a two level systemcoupled to a harmonic oscillator (TLS+HO). Of course, a single harmonic oscillator cannotprovide the 'irreversible' kind of decoherence possible with a bath of many oscillators, but itsufficiently illustrates the ideas behind reduction. The density matrix for the two level systemalone is a Hermitean operator satisfying equations 3-18 through 3-21 above. Such a matrix takesthe most general form

a c+ idc − id b⎛ ⎝ ⎜

⎞ ⎠ ⎟ . 3-27

The two independent diagonal and two independent off-diagonal elements can be constructedfrom the linear combination of the identiy matrix and the set of Pauli matrices

ρ = a+ b2 I + a −b

2 σ z + cσ x + dσ y 3-28where

I =1 00 1⎛ ⎝ ⎜

⎞ ⎠ ⎟ ,σ z =

1 00 −1⎛ ⎝ ⎜

⎞ ⎠ ⎟ ,σ x =

0 11 0⎛ ⎝ ⎜

⎞ ⎠ ⎟ ,σ y =

0 i−i 0⎛ ⎝ ⎜

⎞ ⎠ ⎟ . 3-29

The Pauli matrices will be discussed as representatives for spin 1/2 in the SU(2) group in thechapter on group theory.

Now let the two-level system be coupled to a harmonic oscillator, such that the overallHamiltonian becomes

H = 0 | 0 >< 0 | +E | 1>< 1| +hω (n + 12 ) + V(a

† | 0 ><1 | +a |1 >< 0 |) . 3-30A simple and convenient basis for working with this Hamiltonian is the two degree of freedomproduct basis {|s>|n>}, where s takes on the values 0 or 1 (we could of course call it ±1/2 insteadif we are dealing with a spin 1/2 system) and n takes on the values 0 to infinity. In matrix form,this Hamiltonian can be written as

s = 0 1 0 1 0n = 0 1 2hω / 2 0 0 0 0

hω /2 +E V 0 00 V 3hω / 2 0 00 0 0 3hω / 2+E √2V0 0 0 √ 2V 5hω / 2O

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

3-31

when grouped in blocks of s=0,1. Now consider an initial state of the TLS+HO given by a purestate, i.e. a wavefunction

|Ψ >=12{| s = 0 > + | s = 1 >}× | n = 0 > . 3-32

In this state, the oscillator is in the ground state and the system is in a superposition state. Notethat in addition to the overall wavefunction, the TLS and the HO are also individually describedby wavefunctions 1

2{| s = 0 > + | s = 1 >} and | n = 0 > . The initial density matrix for this state is

ρ =| Ψ >< Ψ | , or in matrix notation in our basis,

3-6

ρ0 =1/2 1/ 2 0 1/2 1/ 2 0 0 0 0O

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ . 3-33

This initial state is particularly simple to propagate because the Hamiltonian in eq. 3-31 is blockdiagonal (one 1× 1 and infinitely many 2× 2 matrices; as seen in the section on group theory,such block-diagonalization in a simple basis is due to a symmetry, here the "dipole-couplingsymmetry"). We can achieve exact propagation of the initial state in 3-33 by considering onlythe upper left 3× 3 block of H and ρ. To further simplify matters, assume that the TLS and HOare in resonance, i.e. E = hω . The diagonal form of the Hamiltonian and eigenvector matrix are

H' =ω / 2 0 0

0 3ω / 2 −V 0−0 0 3ω / 2 +V

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

and M =1 0 00 −1 / 2 1/ 20 1 / 2 1/ 2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ . 3-34

Using this, one easily constructs the diagonal propagator U = exp(−itH / h) , and propagates thedensity matrix ρ0' = M†ρ0M by applying equation 3-8. Finally, ρ' (t ) is transformed back to theoriginal representation using ρ(t) = Mρ' (t)M† . The result is

ρ(t) =

12

14 e

it (ω−V) /h + 14 e

it(ω +V ) /h − 14 e

it(ω −V) /h + 14 e

it(ω +V) /h

14 e

it(V −ω ) /h + 14 e

−it(ω +V )/ h 14 cos(2Vt / h) + 1

4i4 sin(2Vt / h)

− 14 e

it(V −ω) /h + 14 e

−it (ω +V )/ hh − i4 sin(2Vt / h) − 1

4 cos(2Vt / h) + 14

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

. 3-35

One can readily see that the trace equals 1. A somewhat more cumbersome multiplication andtrace show that Tr{ρ2} also equals 1. This is to be expected since we started with a pure stateand propagated with a unitary propagator. But now to the main question: can we still 'split' ρ(t)into separate wavefunctions for the TLS and for the HO? The answer is no.

Evaluating the trace of eq. 3-26 for ρ0 in eq. 3-33 and ρ(t) , one obtains

ρred (0) =

12

12

12

12

⎛⎝⎜

⎞⎠⎟

and ρred (t) =34 −

14 cos(2Vt / ) 1

4 eit (ω −V )/ + 1

4 eit (ω +V )/

14 e

it (V −ω )/ + 14 e

− it (ω +V )/ 14 cos(2Vt / ) + 1

4

⎛⎝⎜

⎞⎠⎟ . 3-36

Both of these are Hermitean matrices with unit trace. The initial matrix also has a unit squaretrace, but for the final matrix

Tr{ρred

2 (t)} = 78+18cos(2Vt / ) 3-37

The TLS oscillates between being in a pure state and a partially dephased state. In the worstcase, for t = πh / 2V, 3-37 drops to 3/4. Unlike at t=0, this reduced density matrix cannot berewritten as a wavefunction: the TLS and the HO are entangled as given by 3-35, and thisentanglement has caused the TLS to dephase. Of course, a single harmonic oscillator cannotmake the system dephase completely; if many harmonic oscillators of different frequencies werecoupled to the TLS, it would loose its coherence entirely at sufficiently long times.

This type of dephasing is also important in quantum measurement theory: macroscopically,one wants to avoid "Schrödinger's cat" states in which the observed system has coherence.Density matrix reduction can explain the absence of such coherence: by observing only the cat,we are averaging over the environment, which causes the cat's coherence to dephase, eliminatingany superposition state. However, the density matrix formalism still cannot predict the outcomeof an individual measurement. A cat that starts out coherently in the left state of eq. 3-36 will beleft in a decohered density matrix with equal probabilities,

12 00 1

2

⎛

⎝ ⎜ ⎞

⎠ ⎟ . 3-38

3-7

3.5 Propagation of the reduced density matrixIn the example above, decoherence does not lead to a complete decay of (ρred)01 because only

a single oscillator is involved. A many-oscillator bath will lead to true equilibration of thesystem reduced density matrix towards the ρeq in eq. 3-25. Below we sketch the derivation of theperturbation expansion leading to an exact expression for the reduced density matrix.

That result can then be used for a simplified treatment of the general propagation problem:the Redfield theory assumes a coupling linear in the bath coordinates (”linear response”),reducing the influence of the bath to a master equation. This leads to exponential dephasing andpopulation decay of the system, as discussed in chapter 11 for a two level system. In manycases, the main effect is as follows: after the system is perturbed from thermal equilibrium, thebath causes diagonal elements of the reduced density matrix to relax back to thermal equilibriumwith an exponential time constant T1, while off-diagonal matrix elements relax back to zero withan exponential time constant T2. One can show that 2T1 > T2. The phase has to relax at acomparable or faster rate than the population.

Let us assume that the Hamiltonian can be split into a time-independent part describing thesystem at equilibrium, such that ρeq = exp[−H0 / kT ] /Tr{exp[−H0 / kT ]} , and a time-dependentpart V(t). To arrive at the equation of motion for ρred, we go to the interaction representation,which for wavefunctions was Ψ I (t) = e

+ iH0 t /Ψ(t) in chapter B: ρI (t) = e

+ iH0 t /ρ(t)e− iH0 t / 3-39Taking the time derivative, inserting equation 3-9 for ρ(t) and canceling terms in H0, theequation of motion is

∂ρI

∂t= −

1i

ρI ,VI[ ] VI = e+ iH0 t /V (t)e− iH0 t / , 3-40

which looks just like the von Neumann equation, except that the dependence on H0 has beenfactored out. For notational convenience, we now introduce the Liouvillian operator L, whichserves as a shorthand for the commutator:

LXρ =

1[ρ,X] 3-41

In terms of L, the equation of motion for ri simply looks like the Schrödinger equation, and wecan integrate it just as in the derivation of time-dependent perturbation theory:

ρI (t) = iLiρI (t)⇒ ρI (t) = ρI (0) − i dt 'LI (t)ρI (t)

0

t

∫ . 3-42

The solutions are obtained by iteration just as in perturbation theory of the wavefunction:

ρI (t) = ρI(n) (t)

n=0

∞

∑ρI(0)(t) = ρeq

ρI(1)(t) = −i dt 'LI (t ')ρeq

0

t

∫

ρI(2)(t) = − dt 'dt ''LI (t ')LI (t '')ρeq

0

t '

∫0

t

∫

3-43

In the last line, the prescription is that we first calculate the commutator of ρ with VI(t’), then thecommutator of the resulting two terms with VI(t”). Now we move from the total density matrix

3-8

to the reduced density matrix. The trace is a linear operation, so we can trace over both sides ofthe eqs. in 3-43. For example,

Trb{ρI(1)(t)} = −i dt 'Trb{LI (t ')ρeq

0

t

∫ }

ρI ,sys(1) (t) = −i dt 'Trb{(ρeq

sys ⊗ ρeqb )VI (t ') −VI (t ')(ρeq

sys ⊗ ρeqb )

0

t

∫ }

= −i dt '(ρeqsysTrb{ρeq

b VI (t ')} − Tr{VI (t ')ρeqb

0

t

∫ }ρeqsys )

= −i dt 'Trb{LI (t ')ρeqb

0

t

∫ }ρeqsys

3-44

The first step is possible because the equilibrium density matrices are diagonal, so they can befactored into a product. The next step takes into account that the trace is only over the bath, andthe last step reassembles the terms into the Liouville operator notation. We can rewrite this bydefining a propagator

U = 1− i dt 'Trb{LI (t ')ρbeq

0

t

∫ } − dt 'dt ''Trb{LI (t ')LI (t '')ρeqb

0

t '

∫0

t

∫ } +

= 1+U (1) + 12!U

(2) +3-45

as

ρIsys (t) =U(t)ρeq

sys ⇒ ρIsys (t) = U(t)ρeq

sys

= U(t)U −1(t)ρIsys (t)

= R(t)ρIsys (t)

3-46

U-1 is not necessarily equal to U† in this case because of the trace in eq. 3-45. Equation 3-46 is afirst order differential equation with variable rate coefficients k ~ -R . If R ( t) were time-independent negative numbers, the solutions would indeed be decaying exponentials. R wouldnever have that form for the full density matrix, but as a result of averaging, R for the reduceddensity matrix is more likely to have all fluctuating terms averaged out and have a negative realpart.