quantum optics in electrical circuits

8/8/2019 Quantum Optics in Electrical Circuits

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QUANTUM OPTICS IN ELECTRIC CIRCUITS

763693S

Erkki Thuneberg

Department of physics

University of Oulu

2010


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Abstract

With present nanofabrication methods it is possible tomake such small electric circuits that quantum effects beco-me essential. The circuits behave like artificial atoms andthe methods to deal with them resemble those used inquantum optics and NMR rather than traditionally usedby electrical engineers. This course is an introduction tothe physics of such circuits. One major topic is how toinclude dissipation into quantum mechanics. This will beanswered by deriving a master equation, and applying itto a harmonic oscillator and to a two-level system. Therealization of the two-level system requires a nonlinear ele-ment, for which superconducting Josephson junctions areused. Another theme is different types of noise (thermal,shot, quantum). These can be derived by applying scat-tering formalism which considers electrons in a conductorlike waves in a transmission line. We try to answer, amongother things, if noise is present at zero temperature, is su-percurrent noisy, and can zero-point fluctuations be mea-sured.

Moderneilla valmistusmenetelmilla voidaan tehda pieni-

kokoisia sahkoisia piireja, joissa kvanttimekaaniset ilmiotovat olennaisia. Nama piirit toimivat kuin keinotekoisetatomit ja niiden kuvaamiseen kaytetaan menetelmia, jot-ka ovat tutumpia kvanttioptiikassa ja ydinmagneettises-sa resonanssissa kuin sahkoopissa. Tama kurssi on joh-datus tallaisten piirien fysiikkaan. Yksi paaaihe on kuin-ka liittaa haviolliset ilmiot kvanttimekaniikkaan. Tamatehdaan johtamalla master-yhtalo, ja sita sovelletaan har-moniseen oskillaattoriin ja kaksitasosysteemiin. Kaksitaso-systeemin toteuttaminen edellytaa epalineaarista element-tia, jona kaytetaan suprajohtavaa Josephson-liitosta. Toi-nen paateema on erityyppiset kohinat kuten lampo-, isku-

ja kvanttikohina. Nama voidaan johtaa kayttaen sironta-formalismia, missa elektroneja johteessa kuvataan kuin aal-toja aaltoputkessa. Tarkoitus on vastata mm. onko nol-lalampotilassa kohinaa, kohiseeko supravirta ja voidaankonollapistevarahtelyja mitata.

Figure: [A. Wallraff et al, Nature 431, 162 (2004)] Inte-grated circuit for cavity QED. a, The superconducting nio-bium coplanar waveguide. b, The capacitive coupling to theinput and output lines. c, False colour electron micrographof a Cooper pair box (blue) fabricated onto the silicon sub-strate (green) into the gap between the centre conductor(top) and the ground plane (bottom) of a resonator (beige)using electron beam lithography and double angle evapo-

ration of aluminium. The Josephson tunnel junctions areformed at the overlap between the long thin island parallelto the centre conductor and the fingers extending from themuch larger reservoir coupled to the ground plane.

Practicalities

Write your name, department/group, year class, andemail address on the list.

The lectures are given in English if there is sufficientdemand for it, otherwise in Finnish.

The web page of the course ishttps://wiki.oulu.fi/display/763693S/Etusivu

The web page contains the lecture material (these notes),the exercises and later also the solutions to the exercises.See the web page also for possible changes in lecture andexercise times.

Time table 2010

Luennot: ti 10-12, sali TE320 (7.9. - 30.11.)

Harjoitukset: ma 10-12, sali TE320 (13.9. - 29.11.)

Teaching assistant: Matti Silveri

Doing exercises is essential for learning. In addition,showing completed exercised will affect your final evalua-tion. (You can improve by one, for example, from 3 to4.) Start calculating (at home) before the exercise time, 2hours is too short time to start from scratch.

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1. Introduction

Content

The following is only an appetizer, and details will beexplained during the course.

harmonic oscillator

electric elements, electric harmonic oscillator, stripline

microwave oscillator, optical cavity, nanomechanical oscil-lator

familiar classical forced and damped harmonic oscillatorsolutions, quadrature variables, rotating wave approxima-tion

= i0 2

+ if. (1)

quantum harmonic oscillator (no dissipation), energy ei-genstates E = 0(n +

12 )

general quantum state |i(t), density operator =

iPi|ii|

general derivation of master equation with dissipativeterms, interaction picture, Markov approximation, rota-ting wave approximation, the master equation for harmonicoscillator

d

dt= i

[H0 + Hd, ]

+

2(N + 1)(2aa aa aa)

+

2N(2aa aa aa) (2)

thermal state solution, coherent states, decaying solution,driving terms, general solutions using translation operator.

for average values, the classical and quantum harmonicoscillators are identical.

statistical aspect, special case of fluctuation-dissipationtheorem

two-state system

qubit = two-state system, how to construct using Jo-sephson junctions, circuit analysis with Josephson junc-tions.

master equation for two level system, rotating wave ap-proximation, Bloch equations.

u = v uT2 ,

v = u vT2

+ Ew,

w = w weqT1

Ev. (3)

harmonic oscillator coupled to a two-state system,Jaynes-Cummings model

Electric transport and noise

transmission lines, transverse modes

temporal modes (wavelets), connection to informationtheory, Landauer formula

G =2e2

hM T (4)

G conductance, M number of transverse modes, T trans-mission probability.

introduction to noise, spectral density SII(),

thermal noise SII() = 2GkT,

shot noise SII() = eI,quantum noise SII() = 2G()

scattering states, derivation of thermal, shot, and quan-tum noise in scattering formalism

Prerequisites

Prior knowledge on quantum mechanics is needed. Themore you know the easier it is, but I try to explain thequantum mechanics that is used.

Books and articles

The first three are standard references in quantum op-tics:

D. F. Walls and G. J. Milburn, Quantum optics (Sprin-ger, Berlin, 1994, second edition 2008). Although not verypedagogical, I have been reading mainly this.

M. O. Scully and M. S. Zubairy, Quantum optics (Cam-bridge, 1997).

C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg,Atom-photon interactions, basic processes and applications(Wiley, New York, 1992).

S. Haroche and J.-M. Raimond, Exploring the quantum- atoms, cavities and photons (Oxford 2006). Contains tho-rough introductions to various topics, experiments with catstates.

L. Allen and J. H. Eberly, Optical resonance and two-level atoms (Dover, New York, 1975). Good account ofBloch equations, but no master equation.

M. Devoret: Quantum fluctuations in electricalcircuits, in Les Houches, Session LXIII, 1995, Quan-tum Fluctuations, (Elsevier 1997). p 351-386. Circuit

theory with Josephson junctions. (download athttp://www.eng.yale.edu/qlab/archives.htm)

S. Datta: Electronic transport in mesoscopic systems(Cambridge, 1995). Explains the scattering formulationand Landauer formula.

Ya. M. Blanter and M. Buttiker: Shot noise in mesosco-pic conductors, Phys. Rep. 336, 1-166 (2000). Scatteringformulation of noise.

C. Beenakker and C. Schonenberger: Quantum Shot Noi-se, Physics Today, May 2003, p. 37-42. A shorter discussionof shot noise.

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Practical applications

No one knows yet

- quantum computation- sensitive measuring devices

Here the main motivation is just to understand the phe-nomena!

Notatione is the charge of an electron. I have tried to formulate

everything so that it is irrelevant whether this quantity isnegative or positive.

= h/2 Plancks constant

a b means that |a| is much smaller than |b|, and cor-respondingly for a b.

a/bc = a/(bc).

The step function

(x) = 1, if x > 0,0, if x < 0.

(5)

cosh x = 12 (ex + ex), sinh x = 12 (e

x ex), tanh x =1/ coth x = sinh x/ cosh x.

2. Harmonic oscillator

Study oscillations around some equilibrium state. If theamplitude of the oscillations is small, nonlinear terms arenot important, one is led to harmonic oscillator.

Examples of harmonic oscillator:

- mass in spring, mechanical pendulum (for small ampli-tudes)- waves in string: each oscillation mode is a harmonic oscil-lator- sound waves: every mode is a harmonic oscillator

electric analog - LC oscillator, modes in a transmissionline, and modes of electromagnetic wave (any frequency:rf, microwave, infrared, light etc) are examples harmonicoscillators.

Figure: [D. Gunnarson et al,Phys. Rev. Lett. 101, 256806 (2008)] LC oscillator coupledto superconducting single-electron transistor. The area ofwhite rectangles in a and b is shown enlarged in b and c,respectively.

Figure: [A. Naik et al, Nature 443, 193 (2006)] Scan-ning electron microscope image of the device: a 21.9-MHz,doubly clamped, SiN and Al nanomechanical resonator(NR) coupled to a superconducting single-electron transis-tor (SSET), with simplified measurement circuit diagram.The SSET is read out by applying 1.17 GHz microwavesthrough the directional coupler, and monitoring the signalreflected off the LCR tank circuit formed by LTank, CTank

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http://dx.doi.org/10.1103/PhysRevLett.101.256806http://dx.doi.org/10.1103/PhysRevLett.101.256806http://dx.doi.org/10.1103/PhysRevLett.101.256806http://dx.doi.org/10.1038/nature05027http://dx.doi.org/10.1038/nature05027http://dx.doi.org/10.1038/nature05027http://dx.doi.org/10.1038/nature05027http://dx.doi.org/10.1103/PhysRevLett.101.256806


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and the SSET.

2.1 Classical harmonic oscillator

Mechanical harmonic oscillator has the equation of mo-tion

p = mq = kq mq + F (6)

with spatial coordinate q, momentum p = mq, mass m,friction constant m, spring constant k, external driving

force F and time derivative marked with a dot. This canbe written in the form

mq + mq + kq = F. (7)

Consider an electric LC oscillator. Electric current I =Q, where Q is the charge. The voltage V can be writtenV = where is the magnetic flux. (This form will beconvenient later, but not necessary at this point.) For thefamiliar electric elements we have the relations: capacitor

Q = CV, inductor = LI, and resistor V = RI. We studyfirst the series oscillator shown in the figure. It is driven bya voltage source (the voltage is independent of the currentflowing through).

C RL

V

For this we deduce

V = i Vi =Q

C+ RQ + LQ. (8)

Its dual, the parallel oscillator, is shown in the followingfigure. It is driven by a current source (the current is inde-pendent of the voltage over the source).

C RLI

For this we find similarly

I = i

Ii

= C +

R+

L. (9)

(Work out the details as an exercise.)

The Maxwell equations

E= 0

,

E= Bt

,

B = 0,B = 00 E

t+ 0j.

The fields E(r, t) and B(r, t) can be represented as sumsof 3 vector components and spatial Fourier components.Consider a single mode, for example

E= E(t)yeikx, B = iB(t)zeikx. (10)

Assume there is some source current j0(r, t), for example,an oscillating charge. It has one Fourier component

j0(t)yeikx that couples to the mode above. In addition we

assume some conductivity of the medium, j = j0 + E.(Alternatively, could consider dielectric loss.) Substitutingto the Maxwell equations we get

00E = kB 0E 0je,B = kE. (11)

This again is of the same form as the other oscillator equa-tions above. Thus a mode of electromagnetic radiation canbe considered as a harmonic oscillator. The whole electro-magnetic field has a huge number of modes which all canbe considered as independent harmonic oscillators.

We notice that all the equations of motion (7), (8), (9),and (11) are of the same form, so we need to study onlyone of them. Although we use the notation of mechanicaloscillator in the following, it should be remembered thatthe analysis is valid for all kinds of harmonic oscillators.

We write k = m20 and

p = m20 q p + Fq =

p

m. (12)

A more symmetric form is

p

m0 = 0m0q p

m0 +F

m0

m0q = 0pm0

. (13)

We define the complex parameters

=12

(

m0q + ip

m0)

=12

(

m0q i pm0

). (14)

Think here simply as some number, there is no quantummechanics involved yet. In these variables the equations ofmotion (12) get an alternative form

= i0 2

+

2 +

iF2m0

(15)

= i0

2 +

2 iF

2m0.

The latter equation is the complex conjugate of the former,and therefore one of them is sufficient. The solution in theabsence of driving and damping is (t) = (0)ei0t, i.e.a rotation in the complex plane with angular frequency0.

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In the following we assume the driving force is harmo-nic, F(t) cos(dt). Note that this is no real restriction,since any function F(t) can be represented as a Fourier se-ries, and because the equation of motion (15) is linear, thesolutions corresponding to each Fourier component can besolved separately, and finally summed together. For nota-tional beauty we also write the prefactor in a convenientform. So we study a driving force of the form

F(t) =

2m0f(t),f(t) = 2f0 cos(dt). (16)

The equations of motion (7) or (15) have an exact so-lution (Exercise). However, later we often need a rotatingwave approximation, and it is appropriate to introduce italready here. The assumption is that the driving and dis-sipative terms are small relative to the first terms on theright hand side of (15). For that we transform to a framerotating at the drive frequency by writing

(t) = (t)eidt. (17)

Substitution gives

= i(0 d) 2

+

2e2idt

+if0(1 + e2idt). (18)

If the driving frequency is near resonance, d 0, thefirst term on the right hand side is small, and supposedly(t) changes slowly in time. In contrast, the two counterrotatingterms contain a rapidly oscillating factor e2idt.The claim is that the effect of these average out on thetime scale we are interested in, which is determined bysecularnon-oscillating terms on the right hand side. The-refore we simply drop out the rapidly oscillating terms.

(The accuracy of the rotating wave approximation could betested by comparison with the exact solution, or by doingperturbation theory with respect of the counter-rotatingterms.) Transforming back to we have the equation ofmotion for a classical harmonic oscillator in the rotatingwave approximation

= i0 2

+ if0eidt. (19)

The equation motion (19) can be solved without dif-ficulties. The steady state solution is obtained by assuming = constant eidt, and one finds

ss(t) = f0eidt

0 d i2 . (20)

The general solution of an initial value problem is

(t) = ss(t) + [(0) ss(0)]ei0t2t

=f0

0 d i2 (eidt ei0t 2 t)

+(0)ei0t2t. (21)

Exercise: plot the real and imaginary part of (t) as afunction of time, and depict the trajectory of (t) in the

complex plane fora) f0 = 0, (0) = 1, 0 = d = 1, = 0.1;b) f0 = 1, (0) = 0, 0 = d = 1, = 0.1, and interpretthe results.

Let us calculate the energy stored in the oscillator as thesum of kinetic and potential energies

E =p2

2m+

1

2m20q

2. (22)

Expressing the energy in terms of (14) gives the verysuggestive form

E = 0||2. (23)Note however that at this point is just some arbitrarynormalization factor that also appears in the definition of (14) and thus it cancels out in the energy (23).

Calculating the time derivative of E (23) and using theequation of motion (19) one gets

dE

dt= Pabs Pdiss,

Pabs = 20(f0eidt),Pdiss = 0||2, (24)

where denotes the imaginary part. Here the first term onthe right hand side can be interpreted as absorption, thepower transferred from the driving force to the oscillator.The latter term, which always is negative, is the dissipation,the power lost in friction.

Let us calculate the absorption and dissipation in thesteady state (20). We find that they are equal, and givenby

Pabs = Pdiss =

0f

2

0(0 d)2 + 14 2 . (25)

One often thinks the absorbed power as function of thedrive frequency, Pabs(d). Its maximum corresponds toresonance, where the drive frequency equals the naturalfrequency of the oscillator, d = 0. The form of Pabs(d)in Eq. (25) is called Lorentzian line shape. Its full width athalf maximum (often abbreviated as FWHM) equals .

It is also instructive to look at

eidtss(t) ss(0) = f00 d i2

=f0(0 d + i2 )(0 d)2 + 14 2

= Aei (26)

in the complex plane. We decompose this complex num-ber into real amplitude A and phase . Here is the pha-se shift between the oscillator spatial coordinate q and thedriving force f. For small driving frequencies 0, i.e. theparticlejust tilts in the direction of the force [as is evi-dent from (7) when the derivative terms can be neglected].With increasing drive frequency grows and gets the va-lue /2 at resonance. It approaches for d , i.e. theparticle tilts in the direction opposite to the force. Apart

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from constant factors, the imaginary part of ss(0) is theabsorption (25). The real part of ss(0) makes the phaseshift to differ from /2 and is called dispersion.

Exercise: plot the absorption (simply [ss(0)]), disper-sion and the phase shift as a function of d, and depictthe trajectory of ss(0) in the complex plane for f0 = 1,0 = 1, = 0.1 and interpret the results.

One often used quantity is the quality factor or Q. It is

a dimensionless number defined as the resonance angularfrequency times the energy stored divided by the dissipatedpower. Using expressions above we calculate

Q =0E

Pdiss=

0

. (27)

The validity of the rotating wave approximation requiresQ 1.

2.2 Quantization

Simple harmonic oscillator

Consider now the harmonic oscillator (12) but withoutdamping and driving,

p = m20 qq =

p

m. (28)

Quantum mechanics makes p and q operators which areHermitian, p = p, q = q. They have to satisfy the com-mutation relations

[q, p] = i, (29)

where [A, B] = AB BA is a commutator. Notice that[A, A] 0 and therefore [q, q] = [p,p] = 0.We make the same change of variables as above (14), but

instead of and we write a and a,

a =12

(

m0q + ip

m0)

a =12

(

m0q i pm0

). (30)

We see that a is the Hermitian conjugate of a as i = i =i. By direct calculation we can now verify that

[a, a

] = 1. (31)

From this relation it follows that the Hilbert space wherea and a operate is spanned by states of the form

|n, n = 0, 1, 2, . . . , , (32)and the following relations determine how a and a operateon these,

aa|n = n|n,a|n = n |n 1,

a|n = n + 1 |n + 1. (33)

Interpretation: aa gives the number ofquanta, a annihila-tes one quantum and a creates one quantum.

For completeness, we give a proof of results (32) and(33).Proof. (read at home) We study the operator

n = aa. (34)

It is Hermitian (verify). The eigenvalues of a Hermitian

operator are real valued. Let us label the eigenstates of theoperator n by using the eigenvalue n (a real number)

n|n = n|n. (35)We assume the eigenstates are normalized, n|n = 1. Wesee that the eigenvalue n cannot be negative,

n = n|n|n = n|aa|n =m

n|a|mm|a|n

=m

|m|a|n|2 0. (36)

We easily calculate

[n, a] = a. (37)

This implies

n(a|n) = an|n a|n = (n 1)(a|n), (38)

and we see that a|n either is a state corresponding to theeigenvalue n1, a|n = c|n1, or else a|n = 0. The lattercase realizes only ifn = 0. In the former case normalizationgives

|c|2 = n|aa|n = n, (39)

and therefore we fix

a|n = n |n 1. (40)

If one operates sufficiently many times with a, one shouldarrive at negative eigenvalues, which is in contradictionwith equation (36). The way out of this is that n is aninteger, so that a|0 = 0, and the process (40) ends. (Notethe essential difference between the n = 0 eigenstate |0and the zero of the linear space 0.) Correspondingly onecan deduce results for a, and we get the results (33).

The energy of the oscillator (22) is the same as the Ha-miltonian of the simple harmonic oscillator

H =p2

2m+

1

2m20 q

2. (41)

Writing it in terms of a and a (14) gives

H = 0

aa +

1

2

. (42)

This allows us to easily find the energy eigenvalues of theoscillator using (33). The eigenstates are |n (32)

H|n = En|n (43)

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and the eigenenergies are

En = 0

n +

1

2

. (44)

We see the energy is quantized to multiples of 0. n isinterpreted as the number of quanta in the oscillator. Thelowest energy state |0 satisfies a|0 = 0. It corresponds tono quanta, n = 0, and zero point energy E = 0/2. The

higher eigenstates can be generated from this by repeatedapplication of a,

|n = (a)nn!

|0. (45)

Above we derived the energy eigenstates of the harmo-nic oscillator in the abstract form (45). The more usualSchrodinger approach also gives the explicit wave functions

n(q) = q|n. (46)These are hardly needed in this course, but for complete-ness let us construct them as well.

In the Schrodinger approach one constructs explicitrepresentation for the operators q and p so that the for-mer is simply multiplication by q and the latter

p = i ddq

. (47)

Substituting these to a (14) and applying to a|0 = 0 gives

(

m0q +

m0

d

dq)0(q) = 0. (48)

The normalized solution of this is

0(q) =m0

14

expm0

2q2

. (49)

All higher states can be obtained by using (45)

n(q) =1n!

m0

2q

2m0

d

dq

n0(q). (50)

These consists of the exponential (49) multiplied by Her-mite polynomials and normalization constants.

4 2 0 2 4q

m0

1

2

3

4

5

6

E

0

Figure: the parabolic potential energy, the 5 lowest ener-gy eigenvalues (horizontal lines) and the corresponding wa-ve functions (50).

10 5 5 10q

m0

0.4

0.2

0.2

0.4

20

Figure: the wave function 20(q).

Forced harmonic oscillator

We include the driving term by postulating a hamilto-nian

H =

p2

2m +

1

2 m2

0 q2

F(t)q. (51)Here the first two terms are the same as the energy ofundriven oscillator (22), and the last one is a new termdescribing driving. Applying the classical Hamilton equa-tions

p = Hq

, q =H

p(52)

we get the same equations of motion as above (12) exceptthe damping term. This justifies the Hamiltonian (51).

Quantizing p and q, and making the change of variables(14) gives

H = 0

a

a +

1

2 (a + a)f(t). (53)

This form is exact. However, in the following we often makeagain the rotating wave approximation. This is best jus-tified going back to the classical Hamiltonian (since wehave not yet discussed the time dependence in the quan-tum case) by replacing a and a by and . Using again(t) = (t) exp(idt) with slow , we see that the drivingterm in (53) is composed of two rapidly oscillating termsand two secular terms. Dropping the former and changingback to quantized notation gives

H = 0aa +1

2 f0(aeidt + aeidt). (54)We see that the forcing causes transitions between the

eigenstates of the simple harmonic oscillator.

Although we could, we are not interested to work on thisfurther before we also have the damping term.

Including damping

The damping term p in the equations of motion (12)cannot be reproduced by any simple term in a Hamilto-nian. Therefore simple minded quantization of the dampedharmonic oscillator fails.

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To have success, we have to think what friction reallyis. It means coupling of a macroscopic variables to somemicroscopic degrees of freedom. Energy is then transferredfrom the macroscopic variables to the microscopic ones. Ifthe number microscopic degrees of freedom is large andthey are sufficiently disordered, the energy will effectivelynot return back to the macroscopic degrees of freedom.Macroscopically this is seen as dissipation, loss of energy.

In order to describe dissipation in quantum mechanics,the obvious way is to include the microscopic degrees offreedom. In a strict sense this means that we have to knowwhat they are. However, we could guess that the grossfeatures of dissipation are independent of the details ofthe microscopic world. Therefore we could use a model forthem. The simplest model we know is the harmonic oscil-lator.

Thus we construct the Hamiltonian

H = H0 + Hd + Hbath + Hint (55)

H0 = 0(aa +

1

2

) (56)

Hd = (a + a)f(t) (57)Hbath =

i

ibi bi (58)

Hint = (a + a)( + ), =i

gibi. (59)

Here H0 is the Hamiltonian of our preferred harmonicoscillator, Hd is the forcing term (53), Hbath describes aset of microscopic harmonic oscillators, and Hint gives thecoupling between the macroscopic and microscopic oscilla-tors (we neglect the zero point energy). The operators bi

and b

i are the annihilation and creation operators for themicroscopic oscillators with frequencies i and satisfy

[bi, bj ] = ij , [bi, bj ] = [b

i , b

j ] = 0. (60)

i.e. all commutators having two different oscillators (inclu-ding also the preferred one) vanish. The coupling betweenthe oscillators is constructed in analogy with the drivingterm so that instead of a fixed function f(t) the time de-pendence comes from dynamical variables + , whichdepend linearly on bis and b

i s with coupling constants gi.

Such a bilinear coupling keeps the quadratic form of theHamiltonian, which makes it not too difficult to solve.

While (55) should be considered as a model of frictionin the case of a mechanical oscillator, it is more than thatfor an electromagnetic resonator. The preferred oscillatoris a mode in the resonator, and it couples to a continuumof electromagnetic modes outside of the resonator.

The Hamiltonian looks complicated. There is an exactsolution, at least in the absence of forcing: one can trans-form to normal coordinates (remember Analytic mechanicscourse). In these, the system just consists of oscillators thatcan be solved independently. This solution is unsatisfacto-ry for several reasons. 1) the transformation to the normalcoordinates and back is complicated, 2) the model contains

a lot of microscopic details (i, gi) and we would like to getrid of these as much as possible, 3) we want to apply thedamping also to other systems than the harmonic oscilla-tor.

In order to make progress, we study more quantum mec-hanics and introduce the density operator.

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3. Quantum mechanics

3.1 Vector spaces

Let us briefly remind about quantum mechanics usingthe Dirac notation. We already used the notation |n for theharmonic oscillator states. In general we assume a linearvector space formed by elements |a called state vectors orstates. The assumption about linear vector space means

that addition and multiplication by a scalar are defined

|a + |b = |c, |d = |f, (61)

with some elements |c and |d also belonging to the vectorspace, and is any complex number. With vector space werequire the standard rules

|a + |b = |b + |a,(|a + |b) + |c = |a + (|b + |c),

(|d) = ()|d,(

|a

+

|b

) =

|b

+

|a

. (62)

In addition, we define another linear vector space formedby states a| so that corresponding to any ketstate |athere is a brastate a|, and vice versa, satisfying

|a + |b = |c a| + b| = c|,|d = |f d| = f|. (63)

We see that the bra space is a copy of the ket space withthe only difference that complex conjugation appears in themultiplication by a scalar. Using this, we further requirethe vector space to form a Hilbert space, so that there existinner product of two state vectors

a

|and

|b

and denoted

by

a|b a||b = (64)

with a complex number . With inner product one requiresthat

a|(|b + |c) = a|b + a|c,a|(|b) = a|b,

a|b = b|a,a|a = 0 |a = 0. (65)

The first two mean linearity with respect to the ket state.Combined with the third these imply linearity also withrespect to the bra state.

Exercise: check that the states

n=0

n|n, (66)

where n are complex and |n (32), form a linear vectorspace. Show that this is Hilbert space if one defines

m|n = m,n. (67)

Let A be a linear operator in the ket space, A|b = |c.The linearity means

A(|b + |c) = A|b + A|c,A(|b) = A|b. (68)

The same operator can be made to operate backwardsinthe bra space, c|A, by requiring

(

c

|A)

|b

=

c

|(A

|b

). (69)

The Hermitian conjugate of A is denoted by A. It has theproperty

b|A|c = c|A|b. (70)The operator A is Hermitian if

b|A|c = c|A|b. (71)Exercise: check that the operators a and a (33) are Her-mitian conjugates and aa is Hermitian.

If

A|b = |b, (72)it is said that |b is an eigenstate of A with eigenvalue . Itcan be shown that for a Hermitian operator all eigenvaluesare real. Two states |b and |c having different eigenvalues = with a Hermitian operator, are orthogonal, c|b = 0.

Let |ek be an orthonormal basis, i.e. any state vectorcan be represented as

|a =k

ck|ek, el|ek = l,k. (73)

The coefficients are then given by ck = ek|a. In the fol-lowing we also assume

|a

is normalized,

a

|a

= 1.

Physical quantities correspond to operators in quantummechanics. Suppose a measurement corresponding to ope-rator B. The possible values obtained in a measurementare the eigenvalues of the operator B. Suppose we se-lect the basis |ek (73) from eigenstates of the operator B.Then the probability to find the eigenvalue is given bythe sum of the squares of the coefficients, |ck|2 of the ei-genstates corresponding to the eigenvalue . The averagevalue, denoted by B, in state |a is given by

B = a|B|a. (74)

Suppose that all eigenvalues of a Hermitian operator are

different i.e. the spectrum is nondegenerate. This allows tolabel the eigenstates by the eigenvalue, |. It also allows torepresent an arbitrary state |a by the set of numbers |a.A special case of this is the wave function a(q) = q|aalready used above (46).

3.2 Time depedence

Based on direct correspondence with classical mechanics,the time dependence in quantum mechanics is given by

idA

dt= [A, H] + i

A

t. (75)

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Here H is the Hamiltonian operator and A =A(q1, q2, . . . , qn, p1, p2, . . . pn, t) is an arbitrary operator,which depends on the canonical operators qi and pi (i =1, . . . , n), and, in some cases, also can depend explicitly ontime, giving rise to the second term on the right hand sideof (75).

As discussed above, the operators are not directly obser-vable quantities, only their eigenvalues. This has the con-sequence that time dependence in quantum mechanics canbe formulated in different ways called pictures so that allobservable quantities are still the same. The equation ofmotion above (75) is valid in Heisenberg picture. In thispicture the state vectors have no time dependence [as all isexpected to be in (75)]. In alternative pictures part of thetime dependence is transferred to the state vectors. Forthat purpose assume a unitary time-dependent operatorU(t). The unitarity means

U U = 1 = UU (76)

with a unity operator 1. If we now define new operatorsand state vectors by

A = UAU, |a = U|a (77)we see that all expectation values are unchanged,

b|A|c = b|A|c. (78)The same applies to the eigenvalues, i.e. there is no physicalchange. However, the equations for the new state vectorsand operators are changed. Now it is most convenient tofirst postulate the following equations in the Schr odingerpicture

id|adt = H|a (79)

idA

dt= i

A

t. (80)

Here the state vectors have time dependence determinedby the Hamiltonian H, but the operators only have theexplicit time dependence. Making the transformation (77)we have

id|a

dt= (UHU + i

dU

dtU)|a

idA

dt= A

, idU

dtU+ U

iA

tU. (81)

Let us fix U(t) by the condition

idU

dt= HU, (82)

and by definition U/t = 0 (find better formulation ofthis?). Then (81) reduce to

id|a

dt= 0 (83)

idA

dt= [A, H] + i

A

t. (84)

These are precisely the same as the Heisenberg pictureequations discussed above (75). Thus this transformationalso justifies the Schrodinger picture postulated above.

One more picture is useful in the following. We decom-pose the Hamiltonian

H = H0 + H1 (85)

where H0 usually is some simple, time-independent part

and H1 is more complicated interactionpart ofH. Let usfix U(t) by the simple part only

idU

dt= H0U U = exp( i

H0t). (86)

Then (81) reduce to equations in the interaction picture

id|a

dt= H1|a (87)

idA

dt= [A, H0] + i

A

t(88)

where H0 = H0.

As an example consider a and a taking as H0 the simpleharmonic oscillator Hamiltonian (42). Using (88) we find

a(t) = aei0t, a(t) = aei0t (89)

where the time dependence in a(t) and a(t) denotesinteraction picture. a and a are the time-independentSchrodinger picture operators.

3.3 Density operator

In order to simplify the description of damping, we haveaverage over the environmental degrees of freedom. Howe-

ver, in quantum mechanics we cannot simply take an ave-rage over a wave function (needs justification). Instead, wefirst form a density operator, and this allows taking avera-ges.

The density operator is the operator

=i

pi|aiai|. (90)

Here pi is the probability that the system is in state|ai. The state vectors |ai are assumed to be normalized,ai|ai = 1. Also the probabilities have to be normalized,

ipi = 1. The case that only one state is possible is calledpure state and the density operator reduces to

= |aa|. (91)The nice thing with the density operator is that the expec-tation value of any operator B can be calculated as

B = Tr(B), (92)where the trace Tr means the sum over the basis vectorsof an arbitrary orthonormal basis |ek,

Tr(. . .) =k

ek| . . . |ek. (93)

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In order to show (92) we calculate

Tr(B) =k

i

ek|aipiai|B|ek

=i

piai|B|ai, (94)

which is the expectation value (74) each state being weigh-ted by its probability pi.

A useful property of the trace is that for any operatorsA and B

Tr(AB) = Tr(BA). (95)

Exercise: show that

Tr = 1. (96)

In order to get some insight, let us calculate the densityoperator corresponding to the state |a = a1|e1 + a2|e2in the basis (|e1, |e2). We get the density matrix

=

11 1221 22

=

|a1|2 a1a2a1a2 |a2|2

. (97)

This pure state should be contrasted with a mixtureofstates |e1 and |e2, where the two states appear with clas-sical probabilities p1 and p2. This gives

=

11 1221 22

=

p1 00 p2

. (98)

The essential difference is the presence of off-diagonal com-ponents 12 and 21 in the coherentsuperposition (97).

We wish to have dynamical equation for the densityoperator. Taking the derivative of (90) and using theSchrodinger equation (79) we get

id

dt= [H, ] . (99)

This is known as von Neumann equation. Note that thedensity operator is exceptional that it does not follow thegeneral rule for operators in the Schrodinger picture (80).Rather the equation (99) resembles the Heisenberg-pictureequation (84), but has the opposite sign. In the interactionpicture (87) one gets

id

dt= [H1,

] (100)

H1 = exp(i

H0t)H1 exp( i

H0t).

3.4 Equilibrium statistical mechanics

In equilibrium at temperature T the probability of eachstate is given by the Gibbs distribution

pi =1

ZeEi = 1

ZeH. (101)

Here = 1/kBT and Z is determined by the normalizationcondition

ipi = 1 or (96). Applying this to the simple

harmonic oscillator (42) gives that the average number ofquanta is given by the Bose distribution

n = 1e0 1 N (102)

and the density operator is

=n=0

(1 e0)e0n|nn|

=n=0

1

1 + N

N

1 + N

n|nn|. (103)

(exercise)

3.5 Correlation functions

Sometimes important information can be expressed by acorrelation function

A(t2)B(t1), (104)where two different times appear. This notation implicitlysays that the operators should be expressed in the Heisen-berg picture.

Exercise: Calculate a(t), a(t), and the correlationfunctions a(t)a(0) and a(t)a(0) for simple harmonicoscillator in equilibrium at temperature T and try to in-terpret the result (for T = 0 and T > 0).

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4. Master equation

4.1 Derivation

We are now ready to go back to the model of damping(55). Our program is to formulate a dynamical equation forthe density operator of the whole system, solve it assumingweak coupling between the macroscopic and microscopicdegrees of freedom, and then average over the microscopic

ones. This should lead to a master equationfor the densityoperator of the macroscopic system only. (WM, section 6.)

For generality, write

H = HS + HR + V (105)

with HS denoting system, HR the reservoir and V theircoupling. Let w(t) denote the density operator of thecomplete system. In the interaction picture it satisfies

idw(t)

dt= [V(t), w(t)] . (106)

The reduced density opeator of the system is

(t) = TrR[w(t)], (107)

where TrR marks the trace over reservoir degrees of free-dom only. We assume that at initial time t = 0 the systemand the reservoir are uncorrelated,

w(0) = (0)R(0). (108)

We integrate (106)

w(t) = w(0) i

t0

dt1 [V(t1), w(t1)] . (109)

Iterating this gives

w(t) = w(0) +n=1

( i

)nt

0

dt1

t10

dt2 . . .

tn10

dtn

[V(t1), [V(t2), . . . [V(tn), w(0)] . . .]]. (110)Taking the trace over the reservoir gives

(t) = (0) +n=1

( i

)nt

0

dt1

t10

dt2 . . .

tn10

dtn

TrR[V(t1), [V(t2), . . . [V(tn), (0)R(0)] . . .]]= (1 + U1(t) + U2(t) + . . .)(0)

= U(t)(0), (111)

where Un(t) is an operator acting on ,

Un(t) = ( i

)nt

0

dt1

t10

dt2 . . .

tn10

dtn

TrR[V(t1), [V(t2), . . . [V(tn), R(0)()] . . .]].We now take the time derivative

d(t)

dt= (U1(t) + U2(t) + . . .)(0)

= (U1(t) + U2(t) + . . .)U1(t)(t)

= l(t)(t), (112)

where the time development operator

l(t) = (U1(t) + U2(t) + . . .)U1(t). (113)

We assume that the interaction term is such that

TrR[V(t)R] = 0. (114)

This implies that U1 = 0. It follows that up to second order

l(t) = U2(t)

= 12

t0

dt1TrR[V(t), [V(t1), R(0)()]],

and thus

d(t)

dt= 1

2

t0

dt1TrR[V(t), [V(t1), R(0)(t)]].

(115)

We now specify to the system-reservoir coupling (59).

First we change to the interaction picture. Secondly, wemake again the rotating wave approximation. It will beseen that the most important case is that all frequenciesare of similar magnitude d 0 i. Thus we can dropterms of type abi and a

bi . We are left with the interaction

V(t) = (a(t)ei0t + a(t)ei0t),(t) =

i

gibieiit. (116)

Substituting into (115) we get

d(t)

dt=

t

0

dt1

TrR

[(a(t)ei0t + a(t)ei0t),

[(a(t1)ei0t1 + a(t1)e

i0t1),

R(0)(t)]]. (117)

Writing out also the double commutator, there will be 16different terms in total. Let us look at one of them,

(1)2t

0

dt1TrR[a(t)ei0tR(0)(t)a

(t1)ei0t1 ]

= a(t)at

0

dt1TrR[(t)R(0)(t1)]e

i0tei0t1

= a(t)at

0dt1TrR[R(0)(t1)(t)]ei

0

(tt1

)

= a(t)aI, (118)

where we have used the cyclic property of trace (95). We seethat the trace is the expectation value (92) in the reservoirand we write

I =

t0

dt1(t1)(t)ei0(tt1)

=

t0

dt1i,j

gigjbibjei(it1+jt)ei0(tt1).

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Thus we have to evaluate the expectation value bibj forthe reservoir oscillators. Assuming they initially are in ther-mal equilibrium, we can use (103) for each oscillator. Alsousing (60) we get

bibj = i,j[1 + N(i)] (119)

I = t

0

dt1i

g2i [1 + N(i)]ei(i0)(tt1).

Assuming the reservoir oscillators have continuous spect-rum with density ()

I =

t0

dt1

0

d1(1)g2(1)[1 + N(1)]

ei(10)(tt1). (120)Changing the integration variables we have

I =

t0

d

0

d1(1)g2(1)

[1 + N(1)]e

i(10)

=

t0

d

0

d(0 + )g2(0 + )

[1 + N(0 + )]ei

=

t0

d

0

df()ei (121)

with f() = (0+)g2(0+)[1+N(0+)]. The integrand

oscillates which leads to cancellations. The main contribu-tion comes from || 1. Assuming t0 1, we can thenextend the integrations to infinity, but simultaneously weadd a convergence parameter > 0 to guarantee that thereis no contribution from =

,

I =

0

d

df()e(i) (122)

and in the end we take the limit 0. Making the timeintegration gives

I = i

df()

+ i(123)

We use the symbolic identity

1

i = P1

i(). (124)

where P denotes the principal value defined by

P

df()

= lima0

a

df()

+

a

df()

. (125)

We get

I = iP

df()

+ f(0)

= i +

2[N(0) + 1] (126)

where

= 2(0)g2(0) (127)

and

= P

d1

(0 + )g

2(0 + )

[1 + N(0 + )]. (128)

Physically is the frequency shift of the system causedby coupling to the reservoir. Usually this is a small effect,and we neglect it here. Later in studying a two-level systemwe see that this shift is similar to the Lamb shift observedin atoms.

One term in the master equation (117) is thus given by(118) and (126).The task that remains is to calculate therest 15 of the 16 terms. Eight of the terms vanish becauseaverages bibj and bi bj vanish in a thermal state (103).The 7 remaining terms can be calculated with slight modi-fications of the above. After that we finally get the masterequation for damped harmonic oscillator in the interaction

picture

d

dt=

2(N + 1)(2aa aa aa)

+

2N(2aa aa aa). (129)

Recall that N is the Bose occupation at the oscillatorfrequency (102), and is a new parameter describing dam-ping, as soon will be seen. If the reservoir is at zero tem-perature, T = 0, we have N = 0 and (129) reduces to

d

dt=

2(2aa aa aa). (130)

If there are other interactionterms in the Hamiltonianof the system, they can be included according to the ge-neral rules of density operator in the interaction picture(100). In particular, the forcing term (57), after a rota-ting wave approximation, can be added to give the masterequation for forced and damped harmonic oscillator in theinteraction picture

d

dt= i

[HdI, ]

+

2(N + 1)(2aa aa aa)

+

2 N(2a

a aa

aa

), (131)HdI = f0[a(t)eidt + a(t)eidt], (132)

with a(t) and a(t) in the interaction picture (89).

Finally, we can also write the master equation in theSchrodinger picture

d

dt= i

[H0 + Hd, ]

+

2(N + 1)(2aa aa aa)

+

2N(2aa aa aa) (133)

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with (56) and (57). More explicity

d

dt= i0[aa, ] + if0[aeidt + aeidt, ]

+

2(N + 1)(2aa aa aa)

+

2N(2aa aa aa). (134)

Let us comment the derivation of the master equation.

We assumed the interaction between system and reservoiras weak. The coupling was taken into account to second or-der in the interaction, which is known as Born approxima-tion. Note that the second order approximation was madein expressing against at the same instant (112). Compa-red to the system at the initial time t = 0 terms of all ordersare included. (Even though the interaction is weak at eachinstant, it can have a huge effect over long time.) Anotherless explicit assumption is called Markovian, the time deri-vative (t) only depends on (t), not any longer history of. Thus the reservoir is assumed to forget any effect causedupon it more rapidly than the system can change its state.We also used the rotating wave approximation. The form of

the damping term in the master equations above is knownas Lindblad form.

4.2 Applications

Averages

As first step learning the master equation, let us calcula-te some expectation values. Consider the annihilation ope-rator,

dadt

=d

dtTr(a). (135)

This can be calculated in any picture and the result is thesame. In the Schrodinger picture a is time independent and

dadt

= Tr(ad

dt). (136)

Substituting from (134) and using (31) and (95) one finds

dadt

= i0a 2a + if0eidt (137)

This equation is precisely the same as the classical equation(19) for . This allows to identify the coefficient in themaster equation as the same damping coefficient as in theclassical theory.

The results for can now be translated to a. Forexample, in the absence of forcing, f0 = 0, we have ex-ponential decay

a(t) = a(0)ei0t 2 t. (138)

Let us apply the same prescription to the number ope-rator n = aa. We get

daadt

= if0(aeidt aeidt)aa + N . (139)

In the absence of forcing the solution is

n(t) = N + (n(0) N)et . (140)

Even if initially n = 0, the occupation will growand approaches exponentially the Bose occupation N =(e0/kBT 1)1. This can be understood that the reser-voir at temperature T > 0 has excitations, and these willexcite the system as well, until a thermal equilibrium is

reached so all oscillators are at the same temperature.

We see that a vanishes in the thermal equilibrium. Thiscan also be seen directly from the equilibrium (103),which only has diagonal elements. This can be unders-tood so that there are quanta present in the oscillator, buttheir phases are random, leading to vanishing a. Clas-sically we can think an ensemble of different k = Ake

ik .If their phases k are random we have = 0 but nonzero||2 = A2, which is the classical quantity correspondingto n.

It is also possible to calculate correlation functions (104).The trick used to calculate correlations like

A(t)B(0), (141)

is called quantum regression theorem. We write the fulltrace of the Heisenberg operators

A(t)B(0) = Tr[U(t)A(0)U(t)B(0)w(0)]= Tr[A(0)U(t)B(0)w(0)U(t)]. (142)

The last form has Schrodinger picture time-independentoperator A(0) and the time-dependent density matrix withthe only difference that B(0) appears at initial time t = 0.In the derivation of the master equation we did not useany special property of the density operator. Thus we cancalculate (142) as above but using the initial value B(0)(0)for the density operator instead of (0). As an example,calculate

da(t)a(0)dt

= i0a(t)a(0) if0a(0)eidt

2a(t)a(0). (143)

The solution in thermal equilibrium, f0 = 0 is

a(t)a(0) = N ei0t 2 |t|, (144)

i.e. thermal equilibrium shows nonvanishing correlationsthat vanish in time exponentially.

Number representation

Let us now study the full density matrix, not only theaverages. One obvious possibility is to represent the densitymatrix in the number basis (32)

=

m=0

n=0

mn|mn|. (145)

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This is an infinite matrix, but often the matrix can betruncated to a finite size with some maximum m and n.The master equation in the number basis is

dmndt

= i0(m n)mn+if0e

idt(

mm1,n

n + 1m,n+1)

+if0eidt(

m + 1m+1,n

nm,n1)

+(N + 1)[

(m + 1)(n + 1)m+1,n+1 12

(m + n)mn]

+N[

mnm1,n1 12

(m + n + 2)mn] (146)

We notice, that only the driving terms contain nondiagonalelements of mn. Assuming f0 = 0 we can thus write theequation for P(n) = nn as

dP(n)

dt= t+(n 1)P(n 1) + t(n + 1)P(n + 1)

[t+(n) + t(n)]P(n) (147)

where

t+(n) = N(n + 1)

t(n) = (N + 1)n (148)

can be interpreted as transition probabilities from state nto states n + 1 and n 1. In the steady state they satisfydetailed balance conditions

t+(n)P(n) = t(n + 1)P(n + 1) (149)

Verify as an exercise that the Bose distribution (103) sa-tisfies the detailed balance condition and thus is a steady

state solution of the master equation (147).

The physical interpretation of the transition probabili-ties will be discussed more later in the connection of atwo-state system.

As one particular solution we can consider the tem-perature relaxation (140) starting from the ground stateP(n) = n,0. Substituting the Bose distribution (103) intothe master equation (147) we see that the solution has theBose distribution at all times with the average number ofquanta satisfying

n(t)

= N(1

et) (150)

and thus corresponding to temperature

T(t) =0kB

1

ln[1 + N1(1 et)1] (151)

with N evaluated at the reservoir temperature.

Glauber states

The following operator is of particular interest

D() = exp(a a), (152)

where is a complex-valued parameter. This is calleddisplacement operator, as will be evident from relation(156) below. Using the relation eA+B = eAeBe[A,B]/2,which is valid if [A, [A, B]] = [B, [A, B]] = 0, the displace-ment operator has the alternative forms

D() = exp(a a)= exp(||2/2) exp(a)exp(a)

= exp(||2

/2) exp(

a) exp(a

). (153)

The exponentials (or any function) of an operator shouldbe understood as Taylor expansions

exp(A) =k=0

kAk

k!. (154)

We verify the properties

D() = D1() = D() (155)

D

()aD() = a + D()aD() = a + . (156)

because [a, f(a)] = f(a), as follows from the commutator(31).

Using the translation operator we construct the followingstates

| = D()|0, (157)These are called Glauber states or coherent states. (Thelatter may be confusing since coherentcan mean also ot-her things.) In order to understand the meaning of thesestates, let us consider the following.

Re

Re

Im Im

Re

Im

|>=D()|0>

|0>

The Hamiltonian of the simple harmonic oscillator hascircular symmetry in the q p variables when expressed inthe variables (23). Since the number states are nondege-nerate, the wave functions (50) are the same when lookedin any coordinates (or ) rotated with respect to byarbitrary angle , (except a possible phase shift). Thus theground state |0 can be represented by a circularly sym-metric dot centered at = 0 of width 1. Now the Glauberstate | is represented by the same dot but displaced by = ||ei. This state is described by a wave function thatis identical to |0 except a coordinate shift, which for the axis would be || cos( ).

Here are some more useful properties of Glauber states.The representation with number states

| = e||2/2n=0

nn!

|n (158)

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and the property that they are eigenstates of the annihila-tion operator

a| = |. (159)The distribution of the photon number in Glauber statesfollows the Poissonian distribution

Pn = |n||2 = e||2 ||2n

n!(160)

with the average and the variance

n = ||2, 2n n2 n2 = n, (161)and standard deviation n =

n. The Glauber statesare not orthogonal,

|||2 = e||2 (162)but are approximately so if | | 1. They satisfy thecompleteness relation

1

||d

2 = 1. (163)

Elimination of driving terms

Let us try the transformation (77) with U = D() on onthe full master equation with the drive (134). We get thatthe effective Hamiltonian corresponding to H = H0 + Hdis [neglecting constants and using (156)]

H = D()HD() + iD()

tD()

= 0(a + )(a + ) f0[(a + )eidt

+(a + )eidt] + i[(

2 a) + ( 2 + a) ]= 0a

a + a[0 f0eidt i]+a[0

f0eidt + i] + constant. (164)The transformed density matrix obeys (we drop tilde)

d

dt= i

[H, ] +

2(N + 1)[2(a + )(a + )

(a + )(a + ) (a + )(a + )]+

2N[2(a + )(a + ) (a + )(a + )

(a + )(a + )]

= i

[H, ] + 2

(N + 1)(2aa aaaa [a, ] + [a, ])+

2N(2aa aa aa + [a, ] [a, ])

= i0[aa, ] + 2

(N + 1)(2aa aa aa)+

2N(2aa aa aa)

i[a, ](0 f0eidt i i 2

)

i[a, ](0 f0eidt + i + i 2

) (165)

Now as long as satisfies the classical equation (19), thelast two terms vanish. This means that the drive completelydrops out of the transformed master equation (165). Basedon this we can now construct several possible solutions forthe driven and damped harmonic oscillator.

1) At T = 0, the equilibrium solution for the transformedmaster equation (165) is the ground state |0. Transformingback to the the original representation, the full solution isa Glauber state

|

(157) with satisfying the classicalequation (19).

2) Starting from an arbitrary initial state but reservoirat T = 0, the transformed master equation (165) givesrelaxation to the ground state, and the solution after initialrelaxation is the Glauber state solution above.

3) At general T, the equilibrium solution for the transfor-med master equation (165) is the thermal equilibrium state(103). The full solution is the thermal state with centerof mass satisfying the classical equation (19).

4) Any thermal relaxation in the transformed masterequation (165) takes place independently of the classical

motion (19) of center of mass. Thus one can have, forexample, initially a coherent state experiencing the thermalrelaxation (150) simultaneously as changes according tothe classical equation (19).

Cat states

Consider a coherent superposition of two Glauber states,

| =N(| + |). (166)Here N is a normalization constant, which is very closeto 1/

2 when | | 1, because of (162). The density

operator corresponding to such a state is

=N2 (|| + || + || + ||) . (167)It is of fundamental interest to study how such an initialstate evolves with time. In particular, the last two termsare off-diagonal and contain information about quantumcoherence between states | and |. That such an inter-ference exists on microscopic states is an essential part ofthe success of the quantum mechanics in explaining phy-sics. However, when applied to macroscopic systems, suchcoherence has not been observed. A famous discussion ofthis was given by Schrodinger, who considered a cat beingeither alive or dead as the two states, and according to nai-

ve extrapolation of quantum mechanics, one could expectto see oscillations between the alive and dead states of thecat. With the Glauber states we can now test this expec-tation since the Glauber states can be macroscopic, thereis no upper limit on the average number of quanta (161).

In order to study this, we first want to verify that

a = ||||e (168)is a solution of the master equation (130) if and satisfythe corresponding classical equation. (For simplicity assu-me T = 0 and f0 = 0.) Here we have used unnormalized

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Glauber states defined by

|| e||2/2| =n=0

nn!

|n (169)

and we need the following properties

a|| = ||, ||a = ||,

a

|| =d

||

d , ||a =d

||d . (170)After some calculation one can now see that (168) indeedsatisfies the master equation. All that remains is to mul-tiply a (168) with an appropriate constant in order toproduce any of the four terms in (167). For the third termwe find the time evolution

3 = N2e 12 (||2+||2|0|2|0|2)e00||= N2e 12 (|0|2200+|0|2)(1et)||= N2e|0|2(1ei)(1et)|| (171)

where 0

and 0

refer to the initial values of and , and inthe last line we have assumed 0 = ei0. For short timesand = the exponential factor is e2nt. For large nthis decays very rapidly. The time constant (2n)1 forthe decay is half of the time constant it takes to emit onequantum to the reservoir. Therefore coherent oscillationsbetween macroscopic states can only take place during avery short time, if at all.

The result above can also be interpreted that the reser-voir is continuously watching the system. Rather than re-servoir, we should speak about the environmentof thesystem, which contains everything except the system itself.Once the system emits one quantum to the environment,

the phase of that photon contains information about thesystem. Thus it acts like a measurement, independently ofwhether any human watches it or not. This kills the cohe-rence between the states, and leaves only the classical pro-bability [the first two terms in (167)]. Which of these twois realized in any particular experiment, can be determinedif the information contained in the emitted photon can berevealed. If not, one has to keep both classical alternatives.

The Glauber states are robust against the effect of theenvironment. Such states are called pointer states, i.e. theycorrespond to some value of a pointer in some classicalmeasuring device. Any coherent superposition of Glauber

states is fragile, i.e. it rapidly decays to a classical proba-bility distribution of different Glauber states. The same inother words, the pointer in the measuring device can haveonly classical probability, it cannot exist in any coherentsuperposition (except of very short times).

Note that also the number states |n are fragile. In thepresence of dissipation a high number state n 1 rapid-ly decomposes to an incoherent probability distribution ofGlauber states | with different phases , where = Aeiand A n.

There are real experiments with the cat states, see HR.

5. Two-state system

In the following we wish to consider the case that theenergy levels are not equally spaced. In this case, one canoften approximate the system by two levels only. A two-state systemmeans that one can excite the system from thelower to the higher level, but any third level is sufficientlyoff resonance that its presence can be neglected. A two-state system is often called a qubit, a short for quantumbit.

In atomic and molecular spectra non-equally spaced le-vels occur as a rule. However, the electric elements we haveconsidered above (capacitors, inductors and resistors) areall linear. Any circuit made of such elements is linear (equi-valently the Hamiltonian is quadratic, at most), and hasequally spaced levels. In order to make a two-state system,one needs a nonlinear element. In the following we study aJosephson junction. This is a tunnel junctionbetween twosuperconducting metals. It is an ideal nonlinear element.The practical drawback is that it works only at low tempe-ratures, where superconductivity appears. In practice thislimitation is not serious because in order to see quantum

effects in electric circuits, one has to cool to low tempera-tures anyway to get rid of noise.

5.1 Circuit analysis

Before going to Josephson junctions, let us exercise a bitcircuit analysis using fluxes and Lagrangian approach. Thegeneral rules are given by Devoret. Here is an attempt toreproduce them.

A circuit consists of branches. To each branch b we as-sociate a potetial difference Vb and a current Ib. To beprecise, these could be defined

Vb(t) =

end of bbeginning of b

E ds,

Ib(t) =1

0

around b

B ds. (172)

About directions we follow the standard but confusing wayof counting Vb positive in the opposite direction as the po-sitive Ib direction. (A more systematic procedure would beto write Ohms law as I = V /R, the potential decreasesin the direction of the current.)

We define branch fluxes and charges by

b(t) =

t

Vb(t)dt

Qb(t) =

t

Ib(t)dt. (173)

The integration is started at some time past where no vol-tages and charges were present. The energy stored in abranch is obtained by calculating

Eb(t) =

t

Vb(t)Ib(t

)dt (174)

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A capacitor is described by the relation Vb = Qb/C. UsingIb = Qb we get the energy

Eb =Q2b2C

=C2b

2. (175)

where we have used Vb = b in the latter form. An inductoris described by the relation b = LIb. Using the samerelations we get the energy

Eb = 2b

2L= LQ

2b

2. (176)

In the following we wish to construct the Lagrangian,which is defined as kinetic energy minus potential energy.In using the flux variables, we can associate capacitive ener-gies as kinetic (because they depend on ) and inductiveenergies as potential (because they depend on ).

A dc voltage source can be represented by adding a fakecapacitor to the circuit, and then taking the limit C and Q while V = Q/C remains constant. Corres-pondingly, a dc current source can be represented by aninductor, and an ac source by an LC circuit.

Not all branch variables are independent variables, theyare constrained by Kirchhoff rules. They say that the vol-tages around each closed loop must sum to zero, and thethe currents arriving to each node must also sum to zero.In terms of flux and charge variables these are

all b around l

b = l (177)

all b arriving at n

Qb = Qn, (178)

where l and Qn are constants. [The node rule (178) is notexplicitly used in the following, it seems to be automatical-ly satisfied by the procedure used.]

C

C3

UL

1

a

b

2

3

Let us consider an example, the circuit in the figure onthe left. The right hand figure describes how we can assignbranch (1, 2 and 3) and node fluxes (ground = 0, aand b) and change the battery to a capacitor.

The loop rule (177) gives 1 + 2 + 3 = . We expressthe branch fluxes in terms of the node ones,

1 = a

2 = b a3 = b. (179)

The next step is to write the Lagrangian as a function ofthe node fluxes. We get

L(a, b, a, b)

=C(b a)2

2+

C32b

2

2a

2L. (180)

We can now take the limit C3 , fixing b = U. Neglec-ting constants we have

L(a, a) = C(a U)2

2

2a

2L. (181)

Using the standard procedure of analytical mechanics, onedefines the canonical momentum corresponding to the nodeflux a as

Qa =L

a= C(a U). (182)

In general this can be identified as the node charge, thesum of the charges on the capacitor plates attached to thenode. Now there is only one such capacitor, which contri-butes the whole Qa. The Hamiltonian is calculated usingthe standard rule

H =

nodes i

iQi L. (183)

In the present case this gives (neglecting constant again)

H = aQa L = C2a

2+

2a2L

. (184)

Writing it as a function of the canonical variables givesfinally

H(a, Qa) =(Qa + CU)

2

2C+

2a2L

. (185)

Except the shift CU caused by the battery, the result isthe same as obtained above (41), taking into account thecorrespondence of the constants in (7) and (9).

5.2 Josephson junctions

In the following we use a simple model of the Joseph-son junction. In superconductivity the electrons form pairs,which are called Cooper pairs. Here we consider only theCooper pairs, and neglect any unpaired electrons. This is areasonable approximation as long as the voltage V is smallcompared to the energy gap of the superconductor

V e

(186)

Often one uses aluminum, which becomes superconducting

below Tc = 1.2 K. The energy gap at T Tc is approxima-tely 0.2 meV (corresponding to 2 K in temperature units,i.e. /kB). Thus we can neglect unpaired electrons forT 1 K and V 0.2 mV. For Nb the conditions aresomewhat more relaxed.

There is another limitation to be encountered soon. Weneed a metallic island that is coupled to the environmentso that it matters if there is one more or less electron onthe island. This means that the capacitive energy of oneelectron Ec = e

2/2C should exceed the thermal energykBT. With nanofabrication methods one can make suchsmall structures that C 1015 F, meaning that T < 1 K.

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We now define a Josephson element by the energy

E = EJcos 20

, (187)

Here 0 = h/2e is the flux quantum and EJ is a cons-tant (coupling energy). Although the cosine form (187)may look strange, a very simple interpretation of this is gi-ven below in connection of equation (199). Using the factthat E = IV = I we can find the current

I = Ic sin2

0= Ic sin . (188)

The latter form follows because we define phase

=2

0 =

2e

. (189)

The relation (188) is known as dc Josephson effect sayingthat at V = 0 there is a constant supercurrentthat dependssinusoidally on (or ). The maximum supercurrent Ic isgiven by

Ic = 2EJ0

= 2eEJ

. (190)

We also see the ac Josephson effect saying that at cons-tant V = 0 there is alternating current as (t) = V t +constant. The frequency of the alternating current is

=2e

V. (191)

At voltage 0.1 mV this gives the frequency = /2 = 48GHz.

5.3 Cooper pair box

C UEJ

Let us now repeat the same circuit calculation as abo-ve but replacing the inductor with a Josephson junction.Going through the calculation one notices that there is notmuch change, and the result is

H(, Q) =(Q + CU)2

2C EJcos

2

0

. (192)

(We have dropped the subindex a as unnecessary.)

UCg

EJ

CJ

=

Let us add one more realistic feature. A real Josephsonjunction always can store charge on the metal electrodes.This can be represented as an additional capacitance CJ

parallel to the Josephson element (figure). Do the analysisfor this circuit diagram (exercise) to find the result

H(, Q) =(Q + Q0)

2

2C EJcos 2

0. (193)

This is the Hamiltonian of a Cooper pair box. We have de-fined Q0 = CgU, C = Cg + CJ. The lead from the batte-ry to the capacitance Cg is called gateand the electrodebetween the Josephson junction and the gate capacitanceis called islandor box.

Until now the circuit analysis has been classical. We canshift to quantum theory using the standard canonical quan-tization procedure. For mechanical system q and p are ca-nonical conjugate variables, and in quantum theory theybecome operators satisfying the commutation relation (29).For the electric circuit considered, the canonical variablesare and Q, and thus analogously they become operatorssatisfying

[, Q] = i. (194)

In Schrodinger picture corresponds to the spatial coor-dinate q (or x) and Q is then the analog of p (47) operator

Q = i dd

. (195)

The Hamiltonian (193) corresponding to the Schrodingereigenvalue equation for the wave function u() is

1

2C

i d

d+ Q0

2u()

EJ cos 20

u() = Eu(). (196)

Alternatively, this can be written using (189)

4Ec

i d

d+

Q02e

2u()EJcos u() = Eu(), (197)

where Ec = e2/2C is the single electron charging energy.

We see that this Schrodinger equation has potential V =EJcos which is periodic.

Before analyzing the solutions of the Schrodinger equa-tion (197), it is useful (if not necessary) to discuss the repre-sentation of the same equation in the charge basis. The ei-genstates of the charge operator (195) are plane waves

Q() = eiQ/

. (198)

Operating with the Cooper pair box Hamiltonian on thisgives

H(, i dd

)Q() (199)

= eiQ/

(Q + Q0)2

2C EJ

2(ei2e/ + ei2e/)

.

The first term is the capacitive energy times the originalcharge eigenstate, as one should expect. The second, Jo-sephson term gives two different charge eigenstates. Their

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charges differ from Q by 2e. Thus we see a simple in-terpretation of the Josephson energy term (187): it trans-fers one Cooper pair across the junction, from side 1 to side2, or vice versa.

Take a look on the circuit diagram of the Cooper pairbox above. In the absence of the Josephson element, theisland is coupled to the rest of the circuit by capacitors only.Thus the charge on the island cannot change. Introducingthe Josephson element changes this situation so that thecharge can be an integral multiple n of 2e, i.e. the possiblevalues of charge

Q = Q1 + 2en (200)

with a constant Q1. Let us denote a charge eigenstate ha-ving n Cooper pairs on the island by |n. Thus the Hamil-tonian (193) can be written in the charge representationas

H =n

4Ec

n +

Q02e

2|nn|

EJ

2 (|n + 1n| + |n 1n|)

. (201)

(Here we have neglected Q1 since it can be eliminated byredefinition of Q0.) Note that in this respect the Joseph-son element is simpler than an inductor. An inductor canpass an arbitrary charge and thus Qa in (185) can varycontinuously and a countable charge basis as in (201) isinsufficient.

What is the meaning of the discrete charge ( 200) in theflux/phase space? The most general allowed wave functionis a discrete superposition of the plane waves (198),

u() =n

unein

(202)

which is now simpler in terms of phase = 2e/ (189)and we neglect Q1. But this condition is the same as

u( + 2) = u() (203)

i.e. the wave function u in (197) has to be periodic and thuscarries independent information only in range < ,say. Note that u and the potential V = EJ cos have thesame period.

It is now convenient to define a new wave function bythe transformation

() = eiQ0/2eu(), ( + 2) = eiQ0/e(). (204)

Substituting into (197) gives

4Ec d2

d2() EJcos () = E() (205)

Thus the dependence on the off-set chargehas movedfrom the Schrodinger equation to the boundary conditions.

The equation (205) has exact solution in terms of Mat-hieu functions. We learn more by considering two limitingcases.

1) Ec EJ, the kinetic energy (of lowest states) is smallcompared to the potential. Close to the minimum at = 0the cosine potential is approximately harmonic. Thus thelowest energy eigenvalues are harmonic oscillator like, witha small correction coming from the 4 term in the Taylorseries of cos(). Q0 has negligible effect on the statessince () 0.

2) Ec EJ, called charge limit. Consider first EJ = 0.As discussed above the solutions in the phase representa-tion are plane waves. The energies are 4Ec(n + Q0/2e)

2.

n=0

0 e 2e 3e 4e

E

Q0

n=1n=-1

EJ

=0

EJ

=/ 0

-e

For any given Q0, the capacitive energy is minimized bythe integer n that is closest to Q

0/2e. This leads to the

set of parabolas as a function of Q0 shown in the figure.The lowest state is nondegenerate except at e, e, 3e etc,where the levels cross. If one now turns on a small EJ,the degeneracy is lifted: the system is in a superposition ofstates where there are either n or n + 1 Cooper pairs onthe island. As depicted in the figure, there are two closelyspaced energy levels, and all others are at much higherenergies. The levels depend essentially on Q0 as there isnot much effect of EJ except near the charge degeneracypoints.

As a summary, we see that a two-state system is for-med in circuits containing a Josephson junction. We have

discussed here so-called charge qubit. Also two other typesof qubits (fluxand phase) can be constructed using Jo-sephson junctions.

5.4 Rabi oscillations

We now consider a general two-state system. We labelthe two levels as a and b, where b is the lower energy state.

|b>

|a>E

The energy separation of the levels is denoted by 0,where 0 0. The general wave function can be written

ab

or | = a|a + b|b. (206)

In the matrix representation the unperturbed Hamiltoniancan be written

H0 =0

2z, (207)

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where we use the Pauli matrices

x =

0 11 0

, y =

0 ii 0

, z =

1 00 1

,

(208)It is also convenient to define

=

0 01 0

, + =

0 10 0

, (209)

These are the same as a and a operating on |1 and |0(33) with the exception that all higher states are cut off,

|a = |b, +|a = 0,|b = 0, +|b = |a. (210)

Let us consider a driven (not damped) qubit. With theanalogy to the harmonic oscillator (53) the Hamiltonian is

H = H0 + Hd =0

2z 2f0x cos dt. (211)

The Schrodinger equationab

= i

2

0 4f0 cos(dt)

4f0 cos(dt) 0

ab

.

(212)We can try to remove the time depedence of H by makingtransformation (77) with

U = exp(idtz/2) =

eidt/2 00 eidt/2

(213)

and get

H =

2 0 d 4f0 cos(dt)eidt

4f0 cos(dt)e

idt

(0

d) .

(214)If we are interested only in a slowly moving solution in thisframe, we may neglect the fast oscillating terms, and get atime independent Hamiltonian

Hrwa =

2

0 d 2f02f0 (0 d)

. (215)

The eigenvalues of this are given by

E = 2

4f20 + (0 d)2. (216)

At resonance (d = 0) the eigenvalues are

f0, and the

solution isab

= A

11

eif0t + B

11

eif0t. (217)

Assuming b as the initial state at t = 0 gives A = B =1/2 and thus a = i sin f0t and b = cos f0t. We find thedensity matrix

=

|a|2 abb

a |b|2

=1

2

1 cos(2f0t) i sin(2f0t)i sin(2f0t) 1 + cos(2f0t)

. (218)

Thus the system oscillates between the ground and excitedstates at the angular frequency R0 = 2f0, known as theRabi frequency. In the general (nonresonant) case thereis oscillation at the (generalized) Rabi frequency R =

4f20 + (0 d)2. Rabi oscillations were first observedby Rabi in atoms (1937). In a Josephson-junction qubitthey were seen by Nakamura et al (1999).

In the following it is useful to introduce a representationof the density matrix in terms a fictitious spin vector S.Any 2 2 matrix can be written in terms of unit matrixI and the three Pauli matrices (208). Taking into accountthat Tr = 1, we can write it as

=1

2I+ S =

12 + Sz Sx iSy

Sx + iSy12

Sz

. (219)

The inverse of this is

Sx =1

2(ab + ba)

Sy =i

2(ab ba)

Sz =

1

2 (aa bb). (220)Note that S is a classical vector, not an operator. It can berepresented by a point in the Bloch sphere depicted below.

Sx

Sy

Sz

Note that the representation (219) is valid for arbitrarystatistical mixtures. For a pure state the fictitious spin isnormalized to 1/2 corresponding to points on the surfaceof the Bloch sphere. For incoherent mixtures S2x + S

2y +

S2z < 1/4 and they correspond to states inside of the Blochsphere.

Consider now an arbitrary 22 matrix Hamiltonian H =cI+ (/2) . Substituting this and (219) into the vonNeumann equation (99) gives the equation of motion

dS

dt=

S. (221)

Thus the vector is the angular velocity of the fictitiousspin S. We can think as a dimensionless fictitious mag-netic field around which the spin rotates.

It is of interest to interpret the previous Rabi oscillationsolution in this picture. First, the static magnetic field ma-kes the spin to rotate around z by 0. The transformation(213) corresponds to changing to a frame rotating around zby d. If d = 0, the only motion in this frame is aroundthe driving field = 2f0x (in RWA), and thus causesthe Rabi oscillation. If d = 0 there is rotation around = 2f0x + (0 d)z.

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We see that with proper driving fields, the qubit can bebrought to any orientation. This is the purpose in quan-tum engineering. The problem is that there are unwantedcouplings to environment, whose effect we will study next.

5.5 Bloch equations

In order to get the damping terms we resort to the mas-ter equation. The master equation for the two-level sys-

tem density matrix is analogous to the harmonic oscillator(133) with substitutions a , a + and thus

d

dt= i

[H0 + Hd, ]

+

2(N + 1)(2+ + +)

+

2N(2+ + +). (222)

Here H0 + Hd is two-state Hamiltonian (211). The bosonoccupation (102) has to be evaluated at the level separation0, i.e.

N = 1exp(0) 1 =

12

coth 0

2 1

, (223)

where the latter form is sometimes more convenient.

Let us stydy the master equation first without driving.Calculating (222) for the 2 2 matrix componentes we get

aa = (N + 1)aa + N bbab = i0ab (N + 12 )abba = i0ba (N + 12 )babb = (N + 1)aa N bb. (224)

We notice that the diagonal and off-diagonal elements of develop independently. The diagonal elements conservethe normalization aa + bb = 1 because bb = aa. Wealso can give physical interpretation to the three termscontributing to

aa = N aa aa + N bb. (225)

The term N bb corresponds to absorption of a quantumfrom the environment. The term N aa corresponds tostimulated emission of a quantum to the environment. Fi-nally, the term aa corresponds to spontaneous emis-sion of a quantum to the environment. The interpretation

of absorption and emission comes because these processesincrease or decrease the energy of the qubit, and from ener-gy conservation. We notice that absorption and stimulatedemission are possible only ifN = 0, i.e. the environment isat finite temperature, T > 0.

One may ask how can the spontaneous emission takeplace by coupling the qubit to the vacuum only (thinkingthe reservoir oscillators as electromagnetic modes). Onecould interpret that even vacuum is not completely emptysince there are zero-point oscillations, and these could kickthe qubit from the upper to the lower state. (We hope toreturn to this soon.)

Similarly to the case of harmonic oscillator, the equi-librium state of the qubit is determined by the detailedbalance condition

aa = 0 (N + 1)aa = N bb. (226)This leads to the Gibbs distribution of the qubit

aa = e0/kBTbb. (227)

In deriving the master equation we assumed the coupling(59) or V(t) = ( + +)[(t) + (t)] between the sys-tem and the reservoir. In addition, we could also considera coupling of the form V(t) = (t)z. (Correct this, Vshould be Hermitian) The derivation of the master equa-tion in this case is much simpler and one finds on the righthand side of the master equation (222) an additional term

4

[z, [z, ]], (228)

where

= 4

0 dt(t)(0). (229)Adding this to (224) gives

aa = (N + 1)aa + N bbab = i0ab (N + 12 )ab abba = i0ba (N + 12 )ba babb = (N + 1)aa N bb. (230)

We see that there is no effect on the diagonal terms, butthe off-diagonal ones acquire additional damping. This canbe understood that the longitudinal coupling induces fluc-tuations into the level separation 0. Thus different rea-

lizations rotate at different speeds, and lead to diminishingof the average coherence. Note that the essential quantityto this dephasing is fluctuations at 0 (229), whereasfor the mixing terms ( ) the fluctuations at 0are dominant. To avoid such fluctuations is one of themain problems in realizing quantum operations in elect-ric circuits.

Let us now represent the damping terms using the spinrepresentation (219). Combining everything gives

dSxdt

= ( S)x 1T2

Sx

dSydt = ( S)y 1T2 SydSzdt

= ( S)z 1T1

(Sz Sz0). (231)

Here we assume the main static part of equals 0z, and

Sz0 = 12(2N + 1)

= 12

tanh02kT

1

T1= (2N + 1) = coth

02kT

1

T2=

1

2T1+ (232)

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where Sz0 is the equilibrium value of the fictitious spin andT1 and T2 are two relaxation times. Equations (231) areBloch equations that have extensively been used in nuclearmagnetic resonance and in optics.

In the literature the Bloch equations appear in variousnotations. Allen&Eberly write the Bloch equations in therotating frame in the form

u =

v

u

T2,

v = u vT2

+ Ew,

w = w weqT1

Ev. (233)

Here u, v and w are the three components ofS, weq = Sz0, = 0 d, and E= 2f0 is the driving field. The steadystate solution for constant Eis

u = weq (T2)(ET2)1 + (T2)2 + T1T2(E)2 ,

v = weqET2

1 + (T2)2 + T1T2(E)2,

w = weq1 + (T2)2

1 + (T2)2 + T1T2(E)2 . (234)

The last one can also be written as

w weq = weq T1T2(E)2

1 + (T2)2 + T1T2(E)2 . (235)

For small driving these reduce to the same as for a har-monic oscillator. At higher drives one gets saturation,where the absorption approaches a maximum. Another ef-fect is resonance fluorescencethat the emitted radiation

is not at the same frequency as the drive, but has three dif-ferent peaks centered at d and d R0. This spectrumcan be determined by studying a Fourier transform of thecorrelation function (104).

5.6 Fluctuation-dissipation theorem

Let us have an interlude to study in more detail theforce fr exerted by the reservoir on the system. By theanalogy of (57) and (59), this force is the operator fr =+. This force cannot be characterized by its mean valuesince it vanishes. Instead, we can use spectral density. Forlater use we give here full definition of this quantity for an

arbitrary quantity f(t). We first define f = f f, thedeviation off from its mean value. Secondly we define theautocorrelation function Rf of f by

Rf() = f(t)f(t ) (236)(In some cases an additional averge over time t may beneeded.) As f may be an operator, the order of f(t) andf(t) is important. The spectral density of f(t) is definedas

Sf() =

Tw/2Tw/2

d eiRf() (237)

where Tw gives the width of the time window used in theFourier transform.

Returning to fr = + we have

Sfr() =

d eifr(t)fr(t )

=

d eifr()fr(0)

=

d ei()(0) + ()(0)

=

d eii,j

gigj(bibjeii + bi bjeii

=

d eii

g2i [(N + 1)eii + N eii]

=

d ei

0

d1(1)g2(1)

[(N(1) + 1)ei1 + N(1)ei1]= 2(||)g2(||) (238

[(N() + 1)() + N()()],because

d ei = 2() (239)

and we have used the step function (x) (5). We see thatthe frequency in Sfr () can be either negative or positive.Using (102) we can write both parts in the same expressionas

Sfr() =Sgn()()

1 e/kBT . (240)

where Sgn() = /|| and was defined above (127) forpositive frequencies and we extend it symmetrically for ne-gative ones,

() = 2(||)g2(||). (241)Using the form (16) we can write the spectral density forthe unreduced force F in (7). We get

SF() =2m()

1 e/kBT . (242)

This result is known as fluctuation-dissipation theorem. Onthe left hand side SF represents fluctuations or noise, the

random force exerted by the reservoir on the system. Onthe right hand side represent dissipation, the frictionalforce caused by the reservoir on the system. This theoremsays that these are related, one cannot have dissipationwithout fluctuations, if T > 0, and vice versa.

Of special interest is the classical limit of (242), meaningkBT . Expanding the exponential factor we get

SF() = 2kBT m(). (243)

Although we have been using the notation of mechanicaloscillator, we must remember that all results are valid for

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all harmonic oscillators, only the constants have differentnames. In particular, comparing (7) with the electricaloscillators equations (8) or (9), we see that voltage andcurrent fluctuations are related to the resistance

SV() = 2kBT R,

SI() = 2kBT/R. (244)

The physics is that a resistor not only dissipates energy, butalso acts as a source of current and voltage fluctuations.

true

resistor V=RI

V=RI

= SI

=

SV

The famous relation (244) was derived by Nyquist (1928)and was motivated by the experimental results by John-son (1928). [Electrical engineers usually define S different-ly from the present physicists way so that they have afactor 4 instead of 2 in (244).]

As an ideal circuit element, the resistance R of a resistoris independent of frequency. This certainly is an idealiza-tion, but is approximately true for real resistors in a rat-her wide range of frequencies. (At least this is limited by 1/ where 1014 s is the Drude relaxation time ofelectrons in a metal). Thus the noise spectral density (244)is independent of frequency. Such noise is called white. Inparticular, it is also symmetric, SV() = SV(). This isnatural for classical noise since any asymmetry must arisefrom non-commutation of f(t) and f(t

) in (236).

Let us now return back to the general expression (242),or equivalently

SV() =2R

1 e/kBT . (245)

What is the meaning of the frequency asymmetry in this expression? For that it is better to go back to the lastform of equation (238). We see that the positive frequencynoise is proportional to N + 1 and the negative one to N.Since there is no other sources of these factors in the deri-

vation of the master equation, we conclude that the termsproportional to N + 1 and N in the master equation arecaused by positive and negative frequency fluctuations. Re-ferring to our discussion of the physical processes caused bythese terms [in connections of (148) and (225)], we can thusidentify positive frequency fluctuations to cause emission,i.e. energy flow from the system to the environment. Thisis effective only if the system initially is in an excited sta-te. Conversely we identify negative frequency fluctuationsto cause absorption, which mean

quantum optics in electrical circuits

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