aa r X i v : . [ qu a n t - ph ] O c t An Introduction to Quantum Computing.
Zachary BurellOctober 25, 2012
Physics has often progressed very rapidly as the precision of measurementshas increased. For instance, it was the precise measurements of Tycho Brahewhich were instrumental in Kepler’s deduction of the elliptic orbit, a resultwhich later formed a cornerstone of Newton’s universal law of gravitation.There is a minimum level of precision in the measurement of the planetaryorbits, below which it becomes impossible to distinguish between elliptical or-bits with minuscule eccentricity, and circular orbits. Once data of the requisiteprecision to notice a difference was available, all that was left for someone todo is to put the pieces together, and voila you get Newton!Quantum Optics, plays an essential role in quantum metrology, a field inwhich the level of precision has increased exponentially over the past twodecades. New techniques of increasing precision in quantum optics have in-creased the significant digits of some of the experimentally measured funda-mental constants by orders of magnitude. Quantum optics is so robust that it1lso of immediate use in testing theories of gravity and quantum field theory,for example, the L.I.G.O. collaboration are using a 2 km baseline MichelsonInterferometer to search for gravitational radiation.These unheralded successes are currently pushing into new domains of ex-perimental precision, and we now have more direct access to the deeper layersof nature. Every week new quantum computing components are brought intobeing, by the shear effort of those working in the field. With each new switch,isolation mechanism, algorithm, etc., the goal of scalable robust quantum com-puting becomes more eminent. If we are successful in constructing quantumcomputers, the effect will be more revolutionary than anything before, includ-ing the classical computer and the internet. The vast expanse of Hilbert Spacewill then be in the throes of man.The fields of quantum optics and quantum computing are closely relatedto one another. Very often breakthroughs in quantum optics are implementedin quantum information processing, storage and quantum communication de-vices. For example, two ways in which cavity QED techniques may be used toto perform quantum computations are (from [8])1. Quantum information can be represented by photon states, with atomstrapped in cavities providing the non-linear interactions between pho-tons, necessary for entanglement.2. Quantum information can be represented by atoms in different states,where photons are used to communicate between the different atoms/states.ny realization of these schemes would at some point have to address the prob-lem of precision control of population transfer, as a means to generate singlephotons. Such precision is a per-requisite for realizing any completely quan-tum technology, that is, any technology based on computational componentswhose functionality depends a’priori on quantum non-linearities, an exampleof which is entanglement.
This treatment of the field quantization will closely (but not exactly) followchapter 2 of Gerry et.al., given in [1]. In order that we understand the inter-action of quantized modes of the electromagnetic field with “ atoms ”, (whosedefinition will, for the moment remain general; we will define an atom to beany bound sate of electrons in a potential V (r) .) we must first understand theproperties of the quantized fields themselves. In the following we begin withthe simple case of a single mode field confined to a 1-d cavity. This clearlyrepresents an idealized situation, but we will later generalize to the case of amultimode field in some three dimensional cavity. We begin as always, with the one-dimensional square well, but in the con-text of quantized modes of the electromagnetic field, which will be relevant forur later analysis of quantized modes of optical cavities , etc. One fruitful andinteresting scenario to investigate for our purposes, is the case of a radiationfield confined to a one dimensional cavity free of sources(i.e. there are no cur-rents,charges, or any dielectric media in the cavity), oriented along what wechoose to be the z axis, with perfectly conducting walls at z = 0 and z = l ,therefore the transverse electric field must vanish at the boundary.Recall that in SI units, the source-free Maxwell equations, which our singlemode field must satisfy, are ∇ × E = ∂ B ∂t (2.1) ∇ × B = µ ε ∂ B ∂t (2.2) ∇ · E = 0 (2.3) ∇ · B = 0 (2.4)We will assume that the field is polarized in the x-direction i.e. E ( r , t ) = e x E x ( z, t ) and hence B ( r , t ) = e y B y ( z, t ) . If we identify q ( t ) as the canonicalposition as defined in the Hamiltonian formalism and similarly identify ˙ q ( t ) as the canonical momentum, then the solution for the components is E x ( z, t ) = (cid:18) ω V ε (cid:19) / q ( t ) sin ( kz ) (2.5)and B y ( z, t ) = (cid:16) µ ε k (cid:17) (cid:18) ω V ε (cid:19) / ˙ q ( t ) cos ( kz ) (2.6)here the wave-number k is related to the frequency ω by k = ω/c . Moreover,the boundary conditions on the electric field at the interface of the perfectconductor at z = 0 and z = L, constrain the values of k to be k = (cid:16) nπL (cid:17) , n = 1 , , .. (2.7)and therefore the allowed frequencies are ω = c (cid:16) nπL (cid:17) , n = 1 , , .. (2.8)We can invert the 2 equations giving E x and B y in terms of the canonicalposition and momentum q ( t ) and ˙ q ( t ) , and obtain the expressions for q ( t ) and ˙ q ( t ) in terms of E x and B y , namely q ( t ) = E x ( z, t ) (cid:18) V ε ω (cid:19) / csc ( kz ) (2.9) ˙ q ( t ) = B y ( z, t ) (cid:18) kµ ε (cid:19) (cid:18) V ε ω (cid:19) / sec ( kz ) (2.10)From these expressions, it is apparent that the Hamiltonian for the field is H = 12 ˆ dV (cid:20) ε E ( r , t ) + 1 µ B ( r , t ) (cid:21) (2.11)Now, ε E ( r , t ) = ε E ( r , t ) · E ( r , t ) = ε ( e x · e x ) E x ( z, t ) ε E x ( z, t ) (2.12)and similarly, µ B ( r , t ) = 1 µ B y ( z, t ) (2.13)Therefore H = 12 ˆ dV (cid:20) ε E x ( z, t ) + 1 µ B y ( z, t ) (cid:21) (2.14)From (1) we have ε E x ( z, t ) = 2 ω V q ( t ) sin ( kz ) (2.15)and µ B y ( z, t ) = 2 V p ( t ) cos ( kz ) (2.16)Therefore (6) becomes H = 12 ˆ dVV (cid:2) ω q ( t ) sin ( kz ) + p ( t ) cos ( kz ) (cid:3) (2.17)Since, cos x = 1 + cos 2 x (2.18)and, sin x = 1 − cos 2 x (2.19)we may write the Hamiltonian as H = 12 ˆ dVV (cid:2) ω q ( t ) (1 + cos 2 kz ) + p ( t ) (1 − cos 2 kz ) (cid:3) (2.20)ow the cosine terms drop out of because of the periodic boundary conditionsand therefore, H = 12 (cid:0) p + ω q (cid:1) (2.21)and so the system is equivalent to harmonic oscillator with unit mass. ( ˙ q ( t ) = p ( t ) ).Now that we have the canonical momentum and canonical position, is isrelatively easy to quantize the field by replacing the variables H with ˆ H and q ( t ) and ˙ q ( t ) with the hermitean (observable) operators ˆ q and ˆ p , respectively.Moreover, we must require that the observables obey the canonical commuta-tion relation [ ˆ q , ˆ p ] = i ~ ˆ I n × n (2.22)which will write from here on out simply as44 [ ˆ q , ˆ p ] = i ~ (2.23)with the n × n matrix identity operator ˆ I n × n implied. Having promoted ˆ q and ˆ p to operators, we are thereby led to the operators for the electric and magneticfields ˆ E x = (cid:18) ω V ε (cid:19) / ˆ q sin ( kz ) (2.24) ˆ B y = (cid:16) µ ε k (cid:17) (cid:18) ω V ε (cid:19) / ˆ p cos ( kz ) (2.25)nd naturally, the Hamiltonian operator becomes ˆ H = 12 (cid:0) ˆ p + ω ˆ q (cid:1) (2.26)Now we define the non-hermitean creation, ˆ a † , and annihilation, ˆ a, opera-tors as follows[1]: √ ~ ω ˆ a † = ( ω ˆ q − i ˆ p ) (2.27) √ ~ ω ˆ a = ( ω ˆ q + i ˆ p ) (2.28)Defining E = ( ~ ω/V ε ) / (2.29)and B = ( µ /k ) (cid:0) ε ~ ω /V (cid:1) / (2.30)it follows that we can write the operators for the electric and magnetic fieldsas [1]: ˆ E x ( z, t ) = E (cid:0) ˆ a † ( t ) + ˆ a ( t ) (cid:1) sin ( kz ) (2.31) ˆ B y ( z, t ) = i B (cid:0) ˆ a † ( t ) − ˆ a ( t ) (cid:1) cos ( kz ) (2.32)From now on we will suppress hats, ˆ , on operators and just write ˆ a † = a † , ˆ a = a, ˆ E x = E x , etc. The benefit of working with creation and annihilationperators is that we are allowed to utilize the simplicity of their algebra. (cid:2) a , a † (cid:3) = aa † − a † a = 1 (2.33)These commutation relations allow us to write the Hamiltonian operator as[2] H = ~ ω (cid:18) a † a + 12 (cid:19) (2.34)In the Heisenberg representation, a general operator ˆ O will obey Heisenberg’sequation of motion [2] d ˆ Odt = ∂ ˆ Odt + i ~ h H , ˆ O i (2.35)Which in the case that ˆ O does not depend explicitly on the time coordinate,becomes d ˆ Odt = i ~ h H , ˆ O i (2.36)Therefore for the creation and annihilation operators we have the followingtime evolution equations[1] da † dt = iωa † (2.37) dadt = − iωa (2.38)which implies that a † ( t ) = a † (0) e iωt (2.39)nd a ( t ) = a (0) e − iωt (2.40)We may expand e − iωt as e − iωt = 1 − iωt − ω t
2! + i ω t
3! + ... (2.41)which allows us to write a ( t ) as a ( t ) = a (0) (cid:18) − iωt − ω t
2! + i ω t
3! + ... (cid:19) (2.42)A useful combination of operators will be a † a = n , a combination known as thenumber operator. If applied to the n th eigenstate of the Hamiltonian | n i (wewill later come to identify | n i as the n photon state), a † a | n i = n | n i (2.43)this operator gives the value n of the eigenstate occupied. The energy eigen-value problem can then be written as H | n i = ~ ω (cid:18) a † a + 12 (cid:19) | n i (2.44) = ~ ω (cid:18) n + 12 (cid:19) | n i = E n | n i (2.45)herefore E n = ~ ω (cid:18) n + 12 (cid:19) . (2.46)Where E is the ground state energy, since is the lowest value which may betaken by n as can be seen from acting on the state | i with the annihilationoperator a . a | i = 0 (2.47)Since a | n i = √ n | n − i . (2.48)and a † | n i = √ n + 1 | n + 1 i . (2.49)It follows that any arbitrary eigenstate | n i can be written in terms of the vac-uum state as (e.g. [1]) | n i = (cid:0) a † (cid:1) n √ n ! | i . (2.50)The states | n i form a complete basis for the Hamiltonian H , and are orthonor-mal h m | | n i = δ mn (2.51)The non vanishing matrix elements of the creation and annihilation operatorsare h n − | a | n i = √ n (2.52) n + 1 | a † | n i = √ n + 1 (2.53). Recall the operator for the electric field given in (8), ˆ E x ( z, t ) = E (cid:0) ˆ a † ( t ) + ˆ a ( t ) (cid:1) sin ( kz ) . (2.54)The eigenstates of the Hamiltonian | n i , do not form a basis for the operator ˆ E x ( z, t ) . This is implied by the fact that the number operator n = a † a , whichdoes commute with the Hamiltonian, does not commute with the electric fieldoperator E x .To see this, let us first calculate the average field h E x ih E x i = h n | ˆ E x ( z, t ) | n i = E (cid:2) h n | a | n i + h n | a † | n i (cid:3) sin ( kz ) (2.55)which, by (10) and (11), become = E [0 + 0] sin ( kz ) = D ˆ E x E = 0 (2.56)that is, the average field is zero.The energy density of the field E = e x E x is proportional to the mean squareof E x , [1] (cid:10) E x (cid:11) = 2 E sin ( kz ) (cid:18) n + 12 (cid:19) (2.57)he variance is defined as [1] (cid:10) (∆ E x ) (cid:11) = (cid:10) E x (cid:11) − D ˆ E x E (2.58)i.e, its the mean square of the standard deviation, which for the eigenstate | ϕ n i becomes ∆ E x = r h E x i − D ˆ E x E = s E sin ( kz ) (cid:18) n + 12 (cid:19) (2.59)that is, ∆ E x = √ E sin ( kz ) (cid:18) n + 12 (cid:19) / (2.60)It is interesting to note that even for n = 0 we have ∆ E x = √ E sin ( kz ) 1 √ E sin ( kz ) (2.61)these are called the vacuum fluctuations of the field, since they correspond tothe eigenstate of the vacuum | i , the state with zero photons. [1]In the case of the electromagnetic field confined to a 1D cavity, the eigen-states of the Hamiltonian, namely | n i , correspond to states of photon number n . One important fact is that the number operator n = a † a and the electric fieldoperator E x do not commute, [ n , E x ] = E sin ( kz ) (cid:0) a † − a (cid:1) (2.62)he generalized uncertainty relations state that for any two operators A and B satisfying [ A , B ] = C , it follows that the product of the uncertainties of A with that of B obey the inequality ∆ A △ B ≥ |h C i| (2.63)It follows therefore, that the number operator and the electric field obey thefollowing uncertainty relations ∆ n ∆ E x ≥ E | sin ( kz ) | (cid:12)(cid:12)(cid:10) a † − a (cid:11)(cid:12)(cid:12) (2.64)This implies a number-phase uncertainty relation [1] ∆ n △ φ ≥ (2.65)where < φ < π is the phase angle associated with the creation and anni-hilation operators. In fact, it will turn out, that the situation is not actuallyquite that simple. It turns out to be a very slippery task to define a uniquephase operator, and in fact is not possible in general [5] ,[6], [7]. It can beshown, however, that for proper definitions of the phase,namely those givenin [4], that the photon number states | ϕ n i have a uniform phase distribution (∆ φ/ ∆ n ) ≈ constant for ≤ φ ≤ π . For more on the number phase uncertaintyrelations see [3], [4] and the references given in [1] . .3 Multimode fields. In free space in the absence of any sources, the source free Maxwell equa-tions are still valid, (joking but obviously true). We write the electric and mag-netic fields in terms of the vector potential A ( r , t ) which satisfies the waveequation [1] ∇ A − c ∂ A ∂t = 0 (2.66)and we choose the Coulomb gauge condition(which will become useful later on) ∇ · A = 0 (2.67)The electric field is then given by[1] E = − ∂ A ∂t (2.68)and the magnetic field is B = ∇ × A (2.69)As long as L ≫ k , we can model free space as cubic cavity, with sides oflength L , therefore we may impose periodic boundary conditions on the facesof the cube[1]. This allows us to deal with the mathematically simpler caseof having a denumerably infinite set of normal modes, rather than a non-denumerably infinite set of modes [1]. We require plane waves in the x i di-ection, where ( i = 1 , , and ( x = x, x = y, x = z ) , to satisfy the condition e ik xi x i = e ik xi ( x i + L ) (2.70)which leads the following conditions for the direction numbers k x i k x = (cid:18) πL (cid:19) m x (2.71) k y = (cid:18) πL (cid:19) m y (2.72) k z = (cid:18) πL (cid:19) m z (2.73)where m x = m y = m z = 0 , ± , ± , ... (2.74)Now, the wave vector k = ( k x , k y , k z ) = 2 πL ( m x , m y , m z ) (2.75)Moreover, k = k k k = √ k · k = ω k /c . Distinct normal modes of the fields arespecified by distinct sets of integers ( m x , m y , m z ) . Therefore, the total numbersof modes in the interval (∆ m x , ∆ m y , ∆ m z ) is [1] ∆ m = ∆ m x ∆ m y ∆ m z = 2 (cid:18) L π (cid:19) ∆ k x ∆ k y ∆ k z (2.76)taking into account a factor of 2 for the two independent polarizations. In theimit that L → ∞ , △ m → dm and we have ( V = L ) dm = dm x dm y dm z = (cid:18) V π (cid:19) dk x dk y dk z (2.77)going to spherical coordinates this is k = ( k x , k y , k z ) = k (sin θ cos φ, sin θ sin φ, cos θ ) (2.78)and therefore dm = (cid:18) V π (cid:19) k dk d Ω (2.79)or using k = ω k /c we can write this as dm = (cid:18) V π (cid:19) ω k kc dω k d Ω (2.80)Integrating over the solid angle Ω , we obtain dm = Vπ k dk = V ρ k dk (2.81)Where ρ k = k /π . We may also write this for dω k as dm = V ω k π c dω k = V ρ ( ω k ) dω k (2.82)for which ρ ( ω k ) = ω k / (cid:0) π c (cid:1) (2.83) Having “pasted” the cubic grid on our space, we may proceed to expand thevector potential as A ( r , t ) = X k ,s e k s (cid:2) A k s ( t ) e i k · r + A ∗ k s ( t ) e − i k · r (cid:3) (2.84)where A k s ∈ C is the amplitude of the field and e k s ∈ R is a polarization vector[1]. Moreover the sum over k is the sum over the distinct sets of integers ( m x , m y , m z ) , and the sum over s is the sum over the two polarization directions[1], which must obey the orthonormality relations e k s · e k s ′ = δ kk ′ δ ss ′ (2.85)The Coulomb gauge condition requires that k · e k s = 0 , which is known as the transversality condition [1].The wave equation and the Coulomb gauge lead to the following relationsfor the amplitudes A k s : d A k s dt + ω k A k s = 0 (2.86)The solution to this differential equation is A k s ( t ) = A k s e − iω k t (2.87) A k s (0) ≡ A k s ) . Thus, the electric and magnetic fields become [1] E ( r , t ) = i X k ,s ω k e k s (cid:2) A k s e i ( k · r − ω k t ) + A ∗ k s ( t ) e − i ( k · r − ω k t ) (cid:3) (2.88) B ( r , t ) = ic X k ,s ω k (cid:18) k | k | × e k s (cid:19) (cid:2) A k s e i ( k · r − ω k t ) + A ∗ k s ( t ) e − i ( k · r − ω k t ) (cid:3) (2.89)The energy of the field is H = 12 ˆ dV (cid:20) ε E · E + 1 µ B · B (cid:21) (2.90)Now [1], (cid:18) k | k | × e k s (cid:19) · (cid:18) k | k | × e k s ′ (cid:19) = δ ss ′ (2.91)and (cid:18) k | k | × e k s (cid:19) · (cid:18) k | k | × e k s ′ (cid:19) = − e k s · e − k s ′ (2.92)Taking our periodic boundary conditions into account, we have ˆ e ± i ( k − k ′ ) · r dV = δ kk ′ V (2.93)Therefore, the contribution to H from the electric field is ˆ dV [ ε E · E ] = ε V X k ,s ω k A k s A ∗ k s − R (2.94)he contribution from the magnetic field is ˆ dV µ B · B = ε V X k ,s ω k A k s A ∗ k s + R (2.95)where, R = 12 ε V X k ,s ω k e k s · e − k s ′ (cid:2) A k s ( t ) A − k s ′ ( t ) + A ∗ k s ( t ) A ∗− k s ′ ( t ) (cid:3) (2.96)Therefore, the total energy in the field is H = 2 ε V X k ,s ω k A k s ( t ) A ∗ k s ( t ) (2.97)but since A k s ( t ) = A k s e − iω k t , this may be written as H = 2 ε V X k ,s ω k A k s A ∗ k s (2.98) We may quantize the field by introducing the canonical position and mo-mentum operators q k s and p k s , respectively through the definitions A k s = 12 ω k √ ε V [ ω k q k s + ip k s ] (2.99) A ∗ k s = 12 ω k √ ε V [ ω k q k s − ip k s ] (2.100)n which case the Hamiltonian becomes H = 12 X k ,s (cid:0) p k s + ω k q k s (cid:1) (2.101)as it should.The canonical variables obey the canonical commutation relations [ q k s , q k ′ s ′ ] = 0 = [ p k s , p k ′ s ′ ] (2.102) [ q k s , p k ′ s ′ ] = i ~ δ kk ′ δ ss ′ (2.103)Just as we did for the single mode field, we may define the creation and anni-hilation operators for the multimode fields p ~ ω k a k s = ω k q k s + ip k s (2.104) p ~ ω k a † k s = ω k q k s − ip k s (2.105)The creation and annihilation operators obey the following commutationrelations [ a k s , a k ′ s ′ ] = h a † k s , a † k ′ s ′ i = 0 (2.106) h a k s , a † k ′ s ′ i = δ kk ′ δ ss ′ δ (cid:16) k ′ − k (cid:17) (2.107)Just as was the case for the single mode field, the number operator for theode k s is n k s = a † k s a k s , and the Hamiltonian is H = X k ,s ~ ω k (cid:18) a † k s a k s + 12 (cid:19) (2.108) = X k ,s ~ ω k (cid:18) n k s + 12 (cid:19) (2.109)Each mode is independent of all the rest and has the eigenstates | n k s i . Ifwe let j denote the j th mode k j s j , then we may write the n th photon numberstate of the j th mode as |{ n j }i = Y j (cid:16) a † j (cid:17) n j p n j ! (cid:12)(cid:12)(cid:12) ϕ ( j )0 E (2.110)The energy eigenvalue equation is then H |{ n j }i = E |{ n j }i (2.111)where [1] E = X j ~ ω j (cid:18) n j + 12 (cid:19) (2.112)A multimode photon state is the tensor product of all of the individual modenumber states, that is | n , n , n ... i = | n i ⊗ | n i ⊗ | n i ⊗ ... (2.113) = |{ n j }i (2.114)he number states are orthogonal, that is h{ n j ′ }| |{ m j }i = Y j,j ′ δ n j m j δ jj ′ (2.115)The multimode vacuum state is |{ j }i = | i ⊗ | i ⊗ | i ⊗ ... (2.116)The action of the creation and annihilation operators on j th mode of the multi-mode photon number state are given by a j |{ n j }i = √ n j (cid:12)(cid:12)(cid:12)n ( n − j oE (2.117) a † j |{ n j }i = p n j + 1 (cid:12)(cid:12)(cid:12)n ( n + 1) j oE (2.118)Quantization requires that the amplitudes A k s become the operators: ˆ A k s = (cid:18) ~ ω k ε V (cid:19) / a k s (2.119)Which therefore allows us to define a vector potential operator as well as elec-tric and magnetic field operators, which are, respectively, ˆA ( r , t ) = X k ,s (cid:18) ~ ω k ε V (cid:19) / e k s h a k s e i ( k · r − ω k t ) + a † k s e − i ( k · r − ω k t ) i (2.120) E ( r , t ) = i X k ,s (cid:18) ~ ω k ε V (cid:19) / e k s h a k s e i ( k · r − ω k t ) − a † k s e − i ( k · r − ω k t ) i (2.121) ˆB ( r , t ) = ic X k ,s ω k (cid:18) k | k | × e k s (cid:19) (cid:18) ~ ω k ε V (cid:19) / h a k s e i ( k · r − ω k t ) − a † k s e − i ( k · r − ω k t ) i (2.122)where the operators a k s = a k s (0) form a basis of the Heisenberg representation .The time dependent creation and annihilation operators are, respectively a † k s ( t ) = a † k s (0) e iω k t (2.123) a k s ( t ) = a k s (0) e − iω k t (2.124)It can be seen that the magnetic field is weaker than the electric field by a fac-tor of /c , which is what we should expect and is a reassuring sign we haven’tgone off track. The magnetic field couples to the spin magnetic moment of theelectrons which is negligible for the aspects of quantum optics which we willinvestigate.Some interesting things to note are: • A single mode plane wave has electric field components given by ˆE ( r , t ) = i (cid:18) ~ ω ε V (cid:19) / e x (cid:2) ae i ( k · r − ωt ) − a † e − i ( k · r − ωt ) (cid:3) (2.125) • Quantum optics often works in the domain of optical radiation, whoseavelength λ is on the order Å.Therefore in these situations we mayapproximate e ± i k · r ≈ ± i k · r (2.126)since, in these situations it is true that λ π = 1 | k | ≫ | r atom | (2.127)and hence the electric field can be expanded as ˆE ( r , t ) ≈ ˆE ( t ) = i (cid:18) ~ ω ε V (cid:19) / e x (cid:2) ae − iωt − a † e iωt (cid:3) (2.128)which is known as the dipole approximation . Consider a single mode field in thermodynamic equilibrium with the wallsof a cavity of absolute temperature T . The density operator ρ for the system is[2] ρ = Z − e − H/kT (3.1)where H is the Hamiltonian, k is Boltzmann’s constant and Z is called the partition function, and it is introduced as a normalization factor, in order thatrace of ρ be one[2]. That is Z = Tr (cid:8) e − H/kT (cid:9) . (3.2)The density matrix for the system is diagonal in the Hamiltonian eigenbasis | ϕ n i since it is in thermodynamic equilibrium. The diagonal matrix compo-nents, gives the population of the stationary state | ϕ n i , which in this case areall the same. They are: ρ th = ρ nn = h n | Z − e − H/kT | n i (3.3) = Z − e − E n /kT . (3.4)Since E n = ~ ω (cid:0) n + (cid:1) , the partition function becomes[1] Z = exp ( − ~ ω/ kT ) X n exp ( − ~ ωn/ kT ) (3.5)Since , exp ( − ~ ω/kT ) < we may sum the series X n exp ( − ~ ω/k B T ) = 11 − exp ( − − ~ ω/kT ) (3.6)Therefore Z = exp ( − ~ ω/kT )1 − exp ( − − ~ ω/kT ) (3.7)The off-diagonal terms vanish, hence there are no coherences between sta-ionary states, and ρ nm = h n | Z − e − H/kT | m i (3.8) = Z − e − E m /kT h n | | m i = 0 . (3.9)We observe that in thermodynamic equilibrium, the populations of the sta-tionary states decrease exponentially with the energy. Since there are no co-herences, the system in this case may be considered to be a statistical mixtureof the states | n i .The probability that the thermal mode is in the n th thermally excited stateis P n = h n | ρ th | n i (3.10) = exp ( − E n /kT ) P n exp ( − E n /kT ) (3.11)The density operator may be written as[1] ρ th = ∞ X n ′ =0 ∞ X n =0 | n ′ i h n | ρ th | n i h n | (3.12) = 1 Z ∞ X n =0 exp ( − E n /kT ) | n i h n | (3.13) = ∞ X n =0 P n | n i h n | (3.14)he average photon number of the thermal field is [1] n = Tr ( nρ th ) = 1exp ( ~ ω/kT ) − (3.15)from which it follows that for kT ≫ ~ ω → n ≈ kT / ~ ω (3.16)While for ~ ω ≫ kT → n ≈ ~ ω/kT (3.17) The Hamiltonian for a system consisting an electron bound to an atom inthe presence of external fields is H (r , t ) = 12 m [ P + e A (r , t )] − e Φ (r , t ) + V ( r ) (4.1)The gauge invariant electric and magnetic fields are given by E ( r , t ) = −∇ Φ (r , t ) − ∂ A ∂t (4.2) B ( r , t ) = ∇ × A (r , t ) (4.3)auge invariance means that these fields are invariant under the gauge trans-formations Φ ′ (r , t ) = Φ (r , t ) − ∂χ (r , t ) ∂t (4.4) A ′ (r , t ) = A (r , t ) + ∇ χ (r , t ) (4.5)Therefore the equation governing the time evolution of the system in theSchrodinger representation, is the Schrodinger equation H (r , t ) Ψ (r , t ) = i ~ ∂ Ψ (r , t ) ∂t . (4.6)In quantum mechanics all operators are invariant under obey a global U (1) similarity transformation; for some operator A , A ′ = U AU † , where U is someunitary operator. Moreover, all state vectors are invariant under multiplica-tion by a common U . This essentially means that given some quantum me-chanical representation of system with all of its operators, and states, etc.,we may obtain an equivalent description of that same system if we simulta-neously transform all of the states and operators of the theory in the mannerprescribed above. The resulting transformed theory will lead to all of the sameresults as the original theory. The usefulness of this fact is that a particularoperator may take on a more tractable form in the transformed theory. There-fore, we may exploit this fact to simplify the Hamiltonian. It will prove usefulfor us to define the unitary operator R which takes us to another representa-ion Ψ ′ (r , t ) of the eigenstate Ψ (r , t ) , by the action of R on Ψ (r , t ) , namely R Ψ (r , t ) = Ψ ′ (r , t ) . (4.7)The transformed Hamiltonian obeys its own Schrodinger equation H ′ (r , t ) Ψ ′ (r , t ) = i ~ ∂ Ψ ′ (r , t ) ∂t (4.8)where [1] H ′ (r , t ) = RHR † + i ~ ∂R∂t R † (4.9)Choosing R = exp ( − ieχ (r , t ) / ~ ) (4.10)which amounts to choosing the Coulomb gauge, we have H ′ = 12 m h P + e A ′ (r , t ) i − e Φ ′ (r , t ) + V ( r ) (4.11)It is important to note that we will be working in the Coulomb gauge, whichis not relativistically covariant, but for which Φ (r , t ) = 0 and ∇ · A = 0 (the transversality condition), therefore the radiation field is completely deter-mined by the vector potential . If there are no sources near the atom, then A satisfies the homogeneous wave equation ∇ A − c ∂ A ∂t = 0 . (4.12)hose solution has the form A = A e i ( k · r − ω k t ) + A † e − i ( k · r − ω k t ) (4.13)In the Coulomb gauge the radiation field is completely determined by the vec-tor potential as can be seen from the Hamiltonian H (r , t ) = P m + em A · P + e m A + V ( r ) (4.14)The transformed Hamiltonian becomes[1] H ′ = 12 m [ P + e ( A + ∇ χ )] − e ∂χ∂t + V ( r ) (4.15)Since | k | = 2 π/λ , it follows that for | r |∼ a ( Bohr radius) and λ ∼ (optical radiation) , k · r ≪ ,Thus we may invoke the dipole approximationwhich gives the first order interactions and which also implies that on lengthscales ∼ | r | atom , the vector potential is locally uniform, and we may make useof the fact that A (r , t ) ≃ A ( t ) . If we choose our gauge function to be χ (r , t ) = − A · r (4.16)it follows that ∇ χ (r , t ) = − A ( t ) (4.17) ∂χ∂t (r , t ) = − r · ∂ A ∂t = − r · E ( t ) (4.18)hich means that H ′ (r , t ) = P m + V ( r ) + e r · E ( t ) (4.19)we recognize the quantity d = − e r as the electric dipole moment.and we maywrite H ′ (r , t ) = H − d · E ( t ) (4.20) Let us begin with the case of a classical field of frequency ω , given by [1] E ( t ) = E cos ( ωt ) Θ ( t ) (4.21)where Θ ( t ) = t > t < (4.22)just means that the field is turned on at a time t = 0 . We can study the in-teraction of an atom with this field by using perturbation theory. Expandingto first order just amounts to using the dipole approximation, k · r ≪ , whichwe have seen previously is satisfied in the case of atoms interacting with aclassical electromagnetic field.Given an atom, in some initial state | i i , we can expand of the atomic stateof the atom for all t > , in a basis of uncoupled atomic states | k i , which spanhe space of H int , | ψ ( t ) i = X k C k ( t ) e − iE k t/ ~ | k i (4.23)where the amplitudes C k ( t ) are normalized such that X k | C k ( t ) | = 1 . (4.24)Now,working in the Schrodinger picture, the atomic state at time t , | ψ ( t ) i ,must obey the time dependent Schrodinger equation, which is: i ~ ∂ | ψ ( t ) i ∂t = (cid:0) H + H int (cid:1) | ψ ( t ) i (4.25)where, in the dipole approximation, as we know from the last section, H int = − d · E ( t ) . Substituting our expression for | ψ ( t ) i into the Schrodinger equation,and then multiplying from the left by h l | e − iE k t/ ~ ,we ( denoting time derivativeswith dots ˙ ) obtain a set of coupled first order differential equations for theamplitudes ˙ C k ( t ) = − i ~ X k C k ( t ) h l | H int | k i e iω lk t Where, ω lk = ( E l − E k ) / ~ , (4.26)are the transition frequencies between atomic states | l i and | k i . In order thatwe may solve these equations we must also subject them to the condition, thatthe initial atomic state is | i i , which implies that C k (0) = 1 . As the state evolvesn time, the initial state will transfer to other other | f i with a probability givenby P i → f ( t ) = | C f ( t ) | . (4.27)To further simplify the task of solving this set of coupled equations analyti-cally, we expand the amplitudes as a power series in some coupling parameter < λ < (which measures the strength of the interaction relative to scale atwhich our theory breaks down, λ = 1 ). C l ( t ) = C (0) l ( t ) + λC (1) l ( t ) + λ C (2) l ( t ) + ... (4.28)Inserting the expression for C l ( t ) into (IV.1) we obtain a recursion for the n th amplitude ˙ C ( n ) l ( t ) = − i ~ X k C ( n − k ( t ) h l | H intlk | k i e iω lk t (4.29)which leads to a coupled set of equations for all of the C nl ( t ) , which up to secondorder are given by ˙ C (0) l ( t ) = 0 (4.30) ˙ C (1) l ( t ) = − i ~ X k C (0) k ( t ) h l | H intlk | k i e iω lk t (4.31) ˙ C (2) l ( t ) = − i ~ X k C (1) k ( t ) h l | H intlk | k i e iω lk t (4.32)The only surviving terms in the sum are those for k = i . Therefore the firstrder amplitude becomes, upon integrating on time C (1) f ( t ) = − i ~ ˆ t dt ′ H intfi e iω fi t ′ C (0) i ( t ′ ) (4.33)using the recursion relation, inserting this value for C (1) l ( t ) into the equationfor ˙ C (2) l ( t ) and integrating on time enables us to find C (2) l ( t ) , (see [1]) C (2) f ( t ) = − i ~ X l ˆ t dt ′ H intfl ( t ′ ) e iω fl t ′ C (1) l ( t ′ ) (4.34) = (cid:18) − i ~ (cid:19) X l ˆ t dt ′ ˆ t ′ dt ′′ H intfl ( t ′ ) e iω fl t ′ H intli ( t ′′ ) e iω fl t ′′ C (0) l ( t ′′ ) (4.35)The total transition probability as a function of time, for a transition fromstate | i i to a state | f i is: P i → f ( t ) = (cid:12)(cid:12)(cid:12) C (0) f ( t ) + C (1) f ( t ) + C (2) f ( t ) + ... (cid:12)(cid:12)(cid:12) (4.36)We have up to now, neglected taking account of the fact that the dipole mo-ment operator d only has non-vanishing matrix elements for states of oppositeparity. Taking this into account we see that up to first order C (0) i ( t ′ ) = 1 , sothat C (1) f ( t ) = − i ~ ˆ t dt ′ H intf i e iω fi t ′ (4.37) = 12 ~ ( d · E ) fi (cid:16) e i ( ω + ω f ) t − (cid:17) ( ω + ω fi ) − (cid:16) e − i ( ω + ω f ) t − (cid:17) ( ω − ω fi ) (4.38)When the radiation frequency ω is near the atomic transition frequency ω fi , weill have resonance, and in this case we may neglect the first “anti-resonant”term since the second term will clearly dominate. This is the so called rotatingwave approximation. With this in mind, the first order transition probabilitybecomes [1] P i → f ( t ) = (cid:12)(cid:12)(cid:12)(cid:16) ( d · E ) fi (cid:17)(cid:12)(cid:12)(cid:12) ~ sin (∆ t/ (4.39)where we have introduced the notation ∆ = ω − ω fi , which is known as the detuning parameter. Earlier, while working in Heisenberg representation, we found that thequanta of a single mode electric field, in the absence of sources of any kind,were given by ˆE ( t ) = i (cid:18) ~ ω ε V (cid:19) / e (cid:2) ae − iωt − a † e iωt (cid:3) (4.40)switching to the Schrodinger representation, this becomes ˆE = i (cid:18) ~ ω ε V (cid:19) / e (cid:2) a − a † (cid:3) (4.41)The free Hamiltonian is H = H atom + H field (4.42)here H atom = P / m + V ( r ) (4.43)and H field = ~ ωa † a (4.44)are the source free Hamiltonians of the atomic system and the field, respec-tively. We have suppressed the vacuum energy in our expression for H field because it does not contribute to the dynamics. The interaction Hamiltonianis H int = − d · E ( t ) = − i (cid:18) ~ ω ε V (cid:19) / ( d · e ) (cid:0) a − a † (cid:1) (4.45) = d · E (cid:0) a † − a (cid:1) (4.46)where, E = i ( ~ ω/ ε V ) / e (4.47)We have thus quantized the atomic system as well as the field system.If we wish to combine the distinct atom and field systems into one, atom-field system, we must remember that the state space of the atom-field systemwill in general be a linear superposition of the eigenstates of H atom and H field .Consider an atomic system in the initial state | a i . If we combine this atomicsystem with the field system which initially contains n photons, then we willhave the atom-fields system which is initially in the state | i i = | a i | n i . (4.48)ince the interaction Hamiltonian H int is proportional to (cid:0) a † − a (cid:1) , it followsthat for the n th eigenstate | n i , the only non-vanishing matrix elements of H int (in the atom-field eigenbasis) are the following (cid:10) H int (cid:11) = X i =1 , h f i | H int | i i = ( d · E ) ba h b , m | (cid:0) a † − a (cid:1) | a , n i = (4.49) = ( d · E ) ba (cid:16) √ n + 1 δ n,.n +1 − √ nδ n,n − (cid:17) . (4.50)where ( d · E ) ba = h b | d | a i · E (4.51)The quantity h b | d | a i = d ba gives the transition dipole moments between thestates | b i and | a i .Therefore the interaction Hamiltonian couples the n th stateto either the n +1 or n − state. In fact H int induces a transition from the initialstate of the atom-field system | i i to the state | f i = | b i | n − i by absorption ofa photon or to the state | f i = | b i | n + 1 i , by the emission of a photon. Theenergies of these states are [1] | i i = | a i | n i ↔ E i = E a + n ~ ω (4.52) | f i = | b i | n − i ↔ E f = E b + ( n − ~ ω (4.53) | f i = | b i | n + 1 i ↔ E f = E b + ( n + 1) ~ ω (4.54)where E a and E b are the energy eigenvalues of the respective atomic states | a i and | b i .et us compare the results of the semi-classical versus the quantum treat-ment of this problem. In both cases, absorption is forbidden in any state forwhich n = 0 (zero photons in the system), for obvious reasons. However, for thequantum case of emission, even if n = 0 transitions may occur known as spon-taneous emission a phenomenon with no semi-classical analog. In cases where n > , we then speak of the stimulated emission of an additional photon. Inthe classical case, if you start with no field, i.e no photon, then you will neverhave a photon later, but a photon later is almost a certainty in the quantumcase. The matrix elements of the interaction are in the case of adsorption [1] h f | H int | i i = h b, n − | H int | a, n i (4.55) = − ( d · E ) ba √ n (4.56)and for the case of emission, are h f | H int | i i = h b, n + 1 | H int | a, n i (4.57) = ( d · E ) ba √ n + 1 (4.58)where, just as before, ( d · E ) ba = h b | d | a i · E (4.59)Fermi’s golden rule tells us that the rates of emission and absorption areproportional to square modulus of the matrix element coupling initial | i i andfinal states | f i , | f i , which in the case of a single mode ( monochromatic) fieldoupled to an atom, whose final state space is spanned by | a i , | b i , The transi-tion matrix elements are given by: (see [1]) W i → [ f ] = π X [ f ] (cid:12)(cid:12)(cid:12) ( d · E ) fi (cid:12)(cid:12)(cid:12) ~ δ ( ω − ω fi ) . (4.60)Moreover, since, ( d · E ) ba = h b | d | a i · E (4.61)this becomes, W i → [ f ] = π X [ f ] |h b | d | a i · E | ~ δ ( ω − ω fi ) . (4.62)Therefore the ratio of these rates is given by |h f | H int | i i| |h f | H int | i i| = n + 1 n . (4.63)For the time being, let’s just focus on two atomic states | a i and | b i , andignore the rest. This allows us to write the write the state vector as | ψ ( t ) i = C i ( t ) | a i | n i e − iE a t/ ~ e − inωt + C f ( t ) | b i | n − i e − iE b t/ ~ e − i ( n − ωt + C f ( t ) | b i | n + 1 i e − iE b t/ ~ e − i ( n +1) ωt (4.64)since we already said that the initial state was | ψ ( t ) i = | a i | n i , so therefore wehave: C i (0) = 1 (4.65) f (0) = C f (0) = 0 (4.66)We can use perturbation theory just before to obtain the probability ampli-tudes for all times, t > . C (1) f ( t ) = − i ~ ˆ t dt ′ h f | H int | i i e iω f i t ′ C (0) i ( t ′ ) (absorption) (4.67) C (1) f ( t ) = − i ~ ˆ t dt ′ h f | H int | i i e iω f i t ′ C (0) i ( t ′ ) (emission) (4.68)where ω f i = ( E f − E i ) / ~ and ω f i = ( E f − E i ) / ~ . Therefore the probabilitythat the atom under goes a transition to the final state | b i is the sum C (1) f = C (1) f + C (1) f , i.e. C (1) f == i ~ ( d · E ) ab " √ n + 1 (cid:0) e i ( ω + ω ba ) t − (cid:1) ( ω + ω ba ) − √ n (cid:0) e − i ( ω + ω ba ) t − (cid:1) ( ω − ω ba ) (4.69)where ω ba = ( E b − E a ) / ~ . If the initial state | a i happens to be the excited statethen ω ba < . In this case, for radiation frequencies ω ∼ ( − ω ba ) , and we getspontaneous emission. Perturbation theory breaks down in situation where we have a drivingfield of near resonance frequency. This is because the resonance causes largepopulation transfers and the assumption made in perturbation theory that C i ( t ) ≈ , no longer holds. Therefore we must take another approach. Oneuch approach can be understood by noticing that for the case of near reso-nance, most of the population is transferred to the near resonant state, so thatthe other states may be neglected. The two most dominant states remain, andthe resulting system of differential equations are much more tractable.The Jaynes-Cummings
Hamiltonian is (see[1],[10]) H = 12 ~ ω σ + ~ ωa † a + ~ λ (cid:0) σ + a + σ + a † (cid:1) . (4.70)The interaction term is H int = ~ λ (cid:0) σ + a + σ + a † (cid:1) (4.71)and it induces the transitions, | e i | n i ↔ | g i | n + 1 i (4.72)or | e i | n − i ↔ | g i | n i . (4.73)The product states | e i | n i , | g i | n + 1 i , etc. , span what is known the bare basis, they are bare states of the Jaynes-Cummings model. For any given n ,the dynamics of the system is confined to a D space of product states spannedby {| e i | n − i , | g i | n − i , | e i | n i , | g i | n i} . (4.74)e can therefore define general product states for any n | ψ n i = | e i | n i (4.75) | ψ n i = | g i | n + 1 i . (4.76)It follows that h ψ n | | ψ n i = 0 (4.77)The matrix representation of HH ij = h ψ in | H | ψ jn i (4.78)In this basis becomes H ij = nω + ~ ω ~ λ √ n + 1 ~ λ √ n + 1 ( n + 1) ω − ~ ω (4.79)For any given n , the energy eigenvalues are E ± ( n ) = (cid:18) n + 12 (cid:19) ~ ω ± ~ Ω n (∆) (4.80)where ∆ = ( ω − ω ) , is the detuning parameter of the atomic transition fre-quency and the monochromatic field and Ω n (∆) = (cid:2) ∆ + 4 λ ( n + 1) (cid:3) / (4.81)s the damped Rabi oscillation frequency , which in the case of resonance, i.e, ∆ = 0 , becomes Ω n (0) = 2 λ √ n + 1 (4.82)The set of energy eigenstates form what are known as the dressed states, andthese are a linear combination of the bare states which are | n, + i = cos (Φ n / | ψ n i + sin (Φ n / | ψ n i (4.83) | n, −i = − sin (Φ n / | ψ n i + cos (Φ n / | ψ n i (4.84)with Φ n given by Φ n = tan − (cid:18) λ √ n + 1∆ (cid:19) = tan − (cid:18) Ω n (0)∆ (cid:19) (4.85)Moreover, sin (Φ n /
2) = 1 √ (cid:20) Ω n (∆) − ∆Ω n (∆) (cid:21) / (4.86) cos (Φ n /
2) = 1 √ (cid:20) Ω n (∆) + ∆Ω n (∆) (cid:21) / (4.87)The dressed states | n, ±i comprise the Jaynes-Cummings doublet. The ~ Ω n (∆) term splits the energies of the bare states | ψ n i , | ψ n i , an effect known as the dynamic stark shift . In the case of exact resonance ∆ = 0 , the bare statesbecome degenerate, but the dynamic stark shift splitting of the dressed statesendures. In the exact resonance limit the dressed states can be represented inhe basis of bare states as | n, + i = 1 √ | e i | n i + | g i | n + 1 i ) (4.88) | n, −i = 1 √ − | e i | n i + | g i | n + 1 i ) (4.89)To obtain the dynamics in the dressed state basis, let us consider the case ofan atom field system, for which the field is prepared in some superposition ofinitial states as | ψ f (0) i = X n C n | n i (4.90)and for which an atom, initially in the state | e i gets injected into the field.Therefore the initial state of the atom field system is | ψ a f (0) i = | ψ f (0) i | e i (4.91) = X n C n | n i | e i = X n C n | ψ n i . (4.92)Now, from (IV.2) and (IV.3), it follows that | ψ n i = cos (Φ n / | n, + i − sin (Φ n / | n, −i (4.93)therefore the initial state of the atom-field system is | ψ af (0) i = X n C n [cos (Φ n / | n, + i − sin (Φ n / | n, −i ] (4.94)ne nice feature of the dressed states, is that they are stationary states of theHamiltonian, and a consequence, the time evolution of H is | ψ af ( t ) i = exp (cid:18) − i ~ Ht (cid:19) | ψ af (0) i (4.95) = X n C n (cid:2) cos (Φ n / | n, + i e − iE + ( n ) t/ ~ − sin (Φ n / | n, −i e − iE − ( n ) t/ ~ (cid:3) (4.96) In the forgoing we have been working with an “on resonance” approxima-tion, in which the detuning parameter vanishes i.e., when ∆ = 0 . A moreinteresting case is the one in which detuning exists to the extent that directatomic transitions do not occur, but where dispersive interactions between asingle and a cavity field do occur .The effective Hamiltonian in the case of large detuning is given by H eff = ~ χ (cid:2) σ + σ − + a † aσ (cid:3) (4.97)Where, χ = λ / ∆ (4.98)The transition operators are the projections σ + = | e i h g | σ − = | g i h e | (4.99)ote that σ + σ − = | e i h e | (4.100)is the emission projector, and the inversion operator σ is σ = | e i h e | − | g i h g | (4.101)The transition and inversion operators obey the Pauli algebra [ σ + , σ − ] = σ (4.102) [ σ , σ ± ] = 2 σ ± (4.103)Suppose that the initial state of the atom-field system has the configurationof an atom in the ground state and the field in a number state, i.e. | ψ (0) i = | g i | n i (4.104)The time evolved state becomes | ψ ( t ) i = e − iH eff t/ ~ | ψ (0) i = e iχnt | g i | n i . (4.105)Similarly, for the initial conditions | ψ (0) i = | e i | n i (4.106)e get | ψ ( t ) i = e − iH eff t/ ~ | ψ (0) i = e iχ ( n +1) t | e i | n i . (4.107)and nothing happens except the production of unmeasurable phase factors.However, if the initial state is a coherent state of the field, that is in thecase where | ψ (0) i = | g i | α i Coherent initial state (4.108)We obtain | ψ ( t ) i = e − iH eff t/ ~ | ψ (0) i = | g i (cid:12)(cid:12) αe iχt (cid:11) (4.109)Similarly, for the initial state | ψ (0) i = | e i | α i (4.110)we have | ψ ( t ) i = e − iH eff t/ ~ | ψ (0) i = e − iχt | e i (cid:12)(cid:12) αe − iχt (cid:11) (4.111)For either case of the initial coherent field state, the coherent state amplitudegets rotated in phase space by the angle θ = χt . The direction of rotationdepends on which initial states the atom is in.Let us now consider the case of an atom in an initial superposition ofground and excited states, which for simplicity, we assume takes the formof a balanced state with the form: | ψ atom i = 1 √ (cid:0) | g i + e iφ | e i (cid:1) φ ↔ phase (4.112)or an initial state | ψ (0) i = | ψ atom i | α i (4.113)We obtain | ψ ( t ) i = e − iH eff t/ ~ | ψ (0) i = 1 √ (cid:0) | g i (cid:12)(cid:12) αe iχt (cid:11) + e − i ( χt − φ ) | e i (cid:12)(cid:12) αe − iχt (cid:11)(cid:1) (4.114)which is a much more interesting state, since now the atom and field are en-tangled .Taking χt = π/ , for which e iχt = i , and e − iχt = − i , we have the entangled state (cid:12)(cid:12)(cid:12)(cid:12) ψ (cid:18) π χ (cid:19)(cid:29) == 1 √ (cid:0) | g i | iα i − ie iφ | e i |− iα i (cid:1) (4.115)which can be understood by analogy to the Schrodinger’s cat paradox. Withthis analogy, our atomic states correspond to the radioactive atom in the para-dox, and the two phase-separated coherent field states play the role of Schrodinger’scat.Moreover the above entangled state corresponds the entangled state | ψ atom − cat i = 1 √ | atom not decayed i | cat alive i + | atom decayed i | cat dead i ) (4.116)Coherent states differing in phase by π are maximally distinguishable, andthere is effectively no overlap between the states, that is for | α | sufficientlylarge. Very large values of | α | are macroscopically distinguishable , while mod-erate values of | α | , mesoscopically distinguishable . Application of CQED to Quantum InformationProcessing
We start this section with a brief overview the Fabry-Perot cavity. One ofthe most essential components of a Fabry-Perot cavity is a partially silveredmirror, which partially reflects and transmits incidents light E a and E b , whichhas the effect of producing output fields E a ′ and E b ′ , which are related by theunitary transformation: E a ′ E b ′ = √ R √ − R √ − R −√ R E a E b (5.1)where R is the reflectivity of the mirror.A Fabry-Perot (FP) cavity is made from two plane parallel mirrors of reac-tivities R and R , incident upon which is light form outside the cavity E int .Inside the cavity, light bounces back and forth between the two mirrors ac-quiring a phase shift e iφ on each trip. The internal cavity field is E cav = X k E k = √ − RE in e iφ √ R R (5.2)One of the most important things about the Fabry Perot cavity for purposesof CQED is the power in the internal cavity field mode as a function of of theower and frequency of the input field, P cav P in = (cid:12)(cid:12)(cid:12)(cid:12) E cav E in (cid:12)(cid:12)(cid:12)(cid:12) = 1 − R (cid:12)(cid:12) e iφ √ R R (cid:12)(cid:12) (5.3)Frequency selectivity arises because of constructive and destructive inter-ference between the cavity mode and the reflected light front. Another indis-pensable feature is that on resonance, the cavity field achieves a maximumwhich is approximately (1 − R ) − times the incident field . Quantum information can be encoded with single photons in the dual railrepresentation c | i + c | i (5.4)Arbitrary unitary transformations can be applied to such quantum informa-tion using phase shifters, beam splitters, and nonlinear optical Kerr media[1].The single photon representation of a qubit is attractive because it repre-sents the information saturation limit of the electromagnetic field and singlephotons, by today’s standards can be generated relatively easily and more-over and most importantly, arbitrary qubit operations become possible,in general in the dual-rail representation . The difficult part in this ap-proach is making the photon - photon scattering amplitudes large enough forentanglement to occur. In an optical cavity this is commonly implemented withptical nonlinear Kerr media. However, in reality even the best non-linearKerr media are weak, and are unable to provide a cross phase modulation of ◦ between single photon states. It is estimated , that even in the best cases,approximately 50 photons would have to be absorbed for each ◦ cross phasemodulated photon.Despite its drawbacks, the optical quantum computer does provide us withsome insight into the architecture and design of a quantum computer. As-suming we had sufficiently good components available, we could construct anoptical quantum computer, which will be almost entirely comprised of opti-cal interferometers. The quantum information is encoded in both the photonnumber states and the photon phase. Interferometers perform the functionof switching between the two representations. Stability however, becomes amajor issue, and if a massive representation of a qubit is chosen, then sta-ble interferometers would be a challenge to construct because of the relativelyshort scale of the de Broglie wavelengths of the qubits. References [1] Gerry, C.G. and Knight, P.L.
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