Holonomic Quantum Computation with Berry’s Phases by Ivette Fuentes-Guridi Notes for the lectures prepared by Francis-Yan Cyr-Racine and Martin Laforest

August 18, 2004

Contents 1 Plan for the lectures

2

2 Some References 2.1 Lecture 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lecture 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Lecture 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 2 2 3

3 Quantum Mechanics

3

4 Phase in Quantum Mechanics 4.1 Dynamical Phase . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Berry’s Phase . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 5 6

5 Parallel Transport

7

6 Spin 1/2 in a Rotating Magnetic Field 6.1 Berry’s phase calculation . . . . . . . . . . . . . . . . . . . . . 6.2 Experimental realization of spin-1/2 particle Berry’s phase . .

1

8 8 10

7 Berry’s Phase in Quantum Computation 12 7.1 Classical Computation . . . . . . . . . . . . . . . . . . . . . . 12 7.2 Quantum Computation . . . . . . . . . . . . . . . . . . . . . 13 7.3 Geometric Computation . . . . . . . . . . . . . . . . . . . . . 14 8

Adiabatic Holonomies

18

9 Non-adiabatic Berry phase: A Generalization 20 9.1 Cyclic Evolution . . . . . . . . . . . . . . . . . . . . . . . . . 20 9.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1

Plan for the lectures 1. Introduction to Berry’s phase; Example with a spin-1/2 particle in a rotating magnetic field 2. Use of Berry’s phase in quantum computing 3. Generalization of the Berry’s phase to the case of degenerate quantum state (Holonomies) ; Introduction to holonomic quantum computing

2

Some References

2.1

Lecture 1

1. M. Berry, Proc. Roy. Soc. Lond., A 392, 45 (1984) 2. B. Simon, Phys. Rev. Lett., 51, 2167 (1983)

2.2

Lecture 2

1. Berry Phase in Quantum Computation:Ekert et al., J. Mod. Opt., 47 (14-15), 2501 (2000) 2. Holonomies:Wilczek and Zee, Phys. Rev. Lett., 52, 2111 (1984) 3. Holonomic Quantum Computation: Zanardi and Rasetti, Phys. Lett. A, 264, 94 (1999)

2

2.3

Lecture 3

• Non-adiabatic: Aharonov and Anandan, Phys. Rev. Lett., 58,1593 (1987) • Holonomic gates: Unanyan, Shore and Bergmann, Phys. Rev. A, 59, 2910 (1999) • Decoherence in Holonomic Quantum Computation:Fuentes-Guridi, Girelli, Livine, Quant-Ph./0311164.

3

Quantum Mechanics

Any coherent quantum state |ψi are vector in a, finite or infinite, vector space over the complex numbers, called a Hilbert space H. A physical measurement of an operator Aˆ1 on a state |ψi gives the expected value of that operator over that state, i.e. ˆ |ψi := hψ|A|ψi ˆ hAi

(1)

Therefore, for a state |ψ 0 i = eiθ |ψi, where θ ∈ R, we have that ˆ |ψ0 i := hψ 0 |A|ψ ˆ 0i hAi ˆ iθ |ψi = hψ|e−iθ Ae = =

ˆ hψ|A|ψi ˆ |ψi hAi

(2)

Thus, from a measurement point of view, it is not possible to distinguish |ψi and |ψ 0 i. This brings us to consider equivalent classes [|ψi], which are define by o n |φi = eiθ |ψi, θ ∈ R (3) [|ψi] := |φi The set containing all the equivalent classes is called a Ray-Hilbert space. In conclusion, global phases are undetectable from measurements, but, if we consider |ψi = eiθ0 |0i + eiθ1 |1i   = eiθ0 |0i + ei(θ1 −θ0 ) |1i 1

note that Aˆ must be hermitian in order to be an observable, i.e. Aˆ† = Aˆ

3

(4)

Figure 1: Mack-Zehnder interferometer. M is a mirror, BS is a beam-splitter and Φ is a phase gate we see that it is actually possible to detect relative phase between different basis component of a state. As an example of relative phase, take the situation of a Mack-Zehnder interferometer as that in Fig. 1. The unitary transformations are given by ! ! ! 0 1 1 1 1 0 1 M= , BS = √2 , Φ= (5) 1 0 1 −1 0 eiφ Therefore, the evolution of the input state |0i is given by BS

|0i −→ Φ

−→ M

−→ M

−→ BS

−→ =

|0i + |1i √ 2 |0i + eiφ |1i √ 2 |1i + eiφ |0i √ 2 |0i + eiφ |1i √ 2 (|0i − |1i) + eiφ (|0i + |1i) √ 2  1 (1 + eiφ )|0i + (1 − eiφ )|1i 2

4

(6)

Therefore, the probability of measuring |0i is given by |h0|ψi|2 = |1 + eiφ |2 = (1 + eiφ )(1 − e−iφ ) 1 (1 + cos φ) = 2

(7)

so that we can, through experiments, find the value of φ.

4

Phase in Quantum Mechanics

In quantum mechanics, phases can be either - Dynamical - Geometrical

4.1

Dynamical Phase

ˆ we a set of eigenstate {|φn i}, such that Suppose we have an Hamiltonian H ˆ n i = En |φn i H|φ Therefore, for any state |ψ(0)i =

P

n an |φn i,

if we take Schr¨odinger equation

∂ |ψ(t)i ∂t ˆ =⇒ |ψ(t)i = e−iHt |ψ(t)i X =⇒ |ψ(t)i = an e−iEn t |φn i

ˆ −iH|ψ(t)i

(8)

=

(9)

n

ˆ = H(t), ˆ The part e−iEn t is called the dynamical phase If H then we have Rt X ˆ 0 0 |ψ(t)i = an (t)Tˆe−i 0 H(t )dt |φn (t)i (10) n

ˆ where {|φn (t)i} are the instantaneous eigenbasis vector of H(t) and Tˆ is the time ordering operator (in case that instantaneous eigenbasis vector at different time do not commute).

5

4.2

Berry’s Phase

ˆ = H( ˆ ~λ), ~λ ∈ Rn . Now, consider H Definition 1. A manifold is a parameter space embedded in a higher dimensional space. ˆ = H( ˆ B) ~ where B ~ is the magnetic field. If we assume Example. Let H ~ ~ lie on the surface of a sphere. In the |B|=constant, then we know that B ~ = B(φ, ~ ~ ∈ R2 spherical coordinates, we can write B θ) so that B Using that parametric hamiltonian, eq. 10 becomes   Rt X 0 ˆ ~ 0 |ψ(t)i = an (t)Tˆe−i 0 H(λ(t ))dt |φn ~λ(t) i

(11)

n

Definition 2. In the adiabatic approximation, the system start in an eigenˆ is varied slowly. state of the hamiltonian and remaing in an eigenstate if H(t) Using the adiabatic approximation with |ψ(0)i = |φn (0)i, we have that |ψ(t)i = e−iγn (t) e−i

Rt 0

En (t0 )dt0

|φn (t)i

(12)

where e−iγn might be a phase acquire during the adiabatic evolution. If we plug this into Schr¨odinger equation, we find iγ˙n |φn (t)i

=

∂ |φn (t)i ∂t

=⇒ γ˙n = −ihφn (t)|

∂ |φn (t)i ∂t

(13)

If we integrate the above equation over a closed path in the parameter space, we thus defined the Berry’s phase γn as I ∂ γn = −i hφn (t)| |φn (t)idt ∂t Z τ ∂ hφn (t)| |φn (t)idt = −i (14) ∂t t=0     ˆ ~λ(τ ) = H ˆ ~λ(0) where τ is defined such that H

6

Note that   ∂ |φn ~λ(t) i ∂t

=

X ∂ dλi |φn i ∂λi dt i

= =⇒ γn

d~λ ∇~λ |φ(~λ)i · dt I = −i hφ(~λ)|∇~λ |φ(~λ)i · d~λ

(15)

Claim. Berry’s phase is gauge invariant, i.e. it does not depend on the global phase of the initial state Proof. Let |φ0n i = eiχ |φn i I

γn0

= −i

hφ0 (~λ)|∇~λ |φ0 (~λ)id~λ

I

= −i hφ(~λ)|e−iχ ∇~λ (eiχ |φ(~λ)i)d~λ I I ~ ~ ~ = i hφ(λ)|∇~λ |φ(λ)i)dλ + i ∇~λ χd~λ = γn

(16)

using the fundamental theorem of calculus, i.e. I ∇~λ χd~λ = 0

(17)

5

Parallel Transport

In quantum mechanics, for a parallel transport along a geodesic, we demand that, to first order, the state does not change and does not even pick up a phase, i.e. 1

'

hψ(t + δt)|ψ(t)i

'

hψ(t)|ψ(t)i + δthψ(t)|

˙ =⇒ hψ(t)|∇~λ |ψ(t)i~λ = 0

∂ |ψ(t)i ∂t (18)

where ~λ is taken and evolved along a geodesic. Three types of parallel transport can be seen in Fig 2 7

Figure 2: Three type of parallel transport around geodesics

6

Spin 1/2 in a Rotating Magnetic Field

6.1

Berry’s phase calculation

~ The hamiltonian Let’s a spin-1/2 particle be in rotating magnetic field B. is given by ˆ = µ ~ H ~ ·B ~ ·B ~ = µS

(19)

~ is the Pauli-spin vector of the particle. If the field is in the z where S direction, then the hamiltonian can be written as ˆ = H

ω σz 2

(20)

We take the 2 eigens of σz to be σz | ↑i = | ↑i, σz | ↓i = −| ↓i

(21)

~ along the y-axis with a certain First, we slowly rotate the magnetic field B angle θ, then around the z-axis for a complete turn and then back to to the original position by a y-rotation of −θ (see Fig. 3). Starting with the spin ”up” state, we have the following transformations: | ↑i

y−rotation

| ↑0 i = e−i 2 σy | ↑i

z−rotation

| ↑00 i = e−i 2 σz e−i 2 σy | ↑i

−→

−→

θ

φ

8

θ

(22)

Figure 3: Then, as seen in eq. 18, the part of the Hamiltonian corresponding to the y rotations do not contribute to the Berry’s phase, since the path in the parameter space is along a geodesic. Therefore, the only contribution to the Berry’s phase will come from the 2π rotation around the z-axis. Thus, using φ as the azimutal angle and ρ as the polar angle, we can compute I γ↑ = −i h↑00 |∇φ | ↑00 idφ Z 2π Z θ = −i ∇φ,ρ × h↑00 |∇φ | ↑00 idφdρ (23) 0

0

where we have used Stoke’s theorem. Now, ∂ 00 ∂ h↑ | | ↑00 i ∂ρ ∂φ   ρ φ ∂ −i σz i ρ2 σy i φ 2 − h↑ |e e σz e−i 2 σz e−i 2 σy | ↑i 2 ∂ρ ρ ρ i ∂ h↑ |ei 2 σy σz e−i 2 σy | ↑i 2 ∂ρ i ∂ h↑ | (cos ρσz + sin ρσx ) | ↑i 2 ∂ρ i ∂ cos ρ 2 ∂ρ i − sin ρ (24) 2

∇φ,ρ × h↑00 |∇φ | ↑00 i = − = = = = =

9

We thus conclude that Z Z 1 2π θ = sin ρdρ 2 0 0 = π(1 − cos θ) Area = 2

γ↑

(25)

A similar calculation shows that the Berry’s phase acquired by a spin initially in the state | ↓i is given by: γ↓ = −π(1 − cos θ)

6.2

(26)

Experimental realization of spin-1/2 particle Berry’s phase

To realize the experimental implantation of the Berry’s phase, we consider the same system as the one describe above, that is, the spin 1/2 particle in a magnetic field. If we take the magnetic field to be in the z-direction, then the Hamiltonian of the system is given by: ω σz 2

ˆo = H

(27)

Now, as in the preceding section, we first rotate the magnetic field with an angle θ around the y-axis and then we rotate it through an angle φ with respect to the z-axis. These two rotations can be described by the unitary operator: U

φ

θ

= ei 2 σz ei 2 σy

(28)

The new Hamiltonian can be found by evolving eq. 27 using the unitary operator describing the rotation: ˆ = UH ˆoU † H φ

φ

= ei 2 σz ei 2 θy = = =



 φ θ σz e−i 2 σy e−i 2 σz

2 φ ω i φ σz e 2 (cos θσz + sin θσx ) e−i 2 σz 2 ω (cos θσz + sin θ cos φσx + sin θ sin φσy ) 2 ω (sin θ cos φ, sin θ sin φ, cos θ) · ~σ 2 10

(29)

By looking at eq. 25, we see that to determine the Berry’s phase of this system, we need a way to find cos θ. To do so, instead of evolving the initial ˆ o , we consider a new Hamiltonian that takes into account the Hamiltonian H interaction between the spin 1/2 particle and the rotating magnetic field. In term of the raising and lowering operators, this Hamiltonian can be written as:   ˆ 0 = ω σz + αλ e−i(φ+νt) σ+ + ei(φ+νt) σ− (30) H 2 where • α = amplitude of the field • ν = frequency of the field • φ = phase of the field • λ = coupling constant. We can simplify this Hamiltonian by writing it in a frame that rotates at a frequency ν around the z axis:   ˆ 00 = ω − ν σz + +αλ e−iφ σ+ + eiφ σ− H (31) 2 Now taking ∆ = ω − ν and the definition of the raising and lowering operators, we have:     ∆ −iφ σx + iσy iφ σx − iσy 00 ˆ H = σz + αλe + αλe 2 2 2  iφ   iφ   −iφ −iφ ∆ e +e e −e = σz + αλ σx + σy 2 2 2i ∆ = σz + αλ (cos φσx + sin φσy )  2 ∆ = αλ cos φ, αλ sin φ, · ~σ 2 = Bo (sin θ cos φ, sin θ sin φ, cos θ) · ~σ (32) The last step comes from the requirement that this Hamiltonian should be ˆ o with the unitary rotation operators similar to the one found by evolving H

11

(see eq. 29). Now, if we compare the last two lines of the above calculation, we can see that the value of Bo and cos θ that satisfy this equation are: Bo =

1 √ , 2 ∆2 +4λ2 α2

cos θ =



∆ ∆2 +4λ2 α2

(33)

This allows us to experimentally determine the Berry’s phase since we can express it as a function of the magnetic field parameters: γ↑ = π(1 − cos θ)   ∆ = π 1− √ ∆2 + 4λ2 α2

7 7.1

(34)

Berry’s Phase in Quantum Computation Classical Computation

The most fundamental thing in classical computing is the bit. A bit can take two physical distinguishable states given by either ”0” or ”1”. In classical computation, we store the input information in a string of bits, for example ”010011001”. This input state goes into a computational process which can be governed only by the laws of classical physics (interaction algorithm). The output is the solution to the given computational problem. Also, in classical computation, there is a set of universal gates such that any computation algorithm can be made out of them. These gates transform the state of the bits so that we can get a logic output out of them. The Nand is an example of such a gate. Its action is described in figure 4.

Nand Input Output 00 1 10 1 01 1 11 0 Figure 4: The output of the Nand gate for a two bits input.

12

7.2

Quantum Computation

In quantum computation, we encode the information in qubits (quantum bits). The qubits are represented, similarly to the classical case, by two distinguishable quantum states |0i and |1i. What is different in quantum computation is that a qubit can be in a superposition of the two basis states. For example: |ψi = α|0i + β|1i

(35)

such that |α|2 + |β|2 = 1. However, which such qubits, the readout of the output of the computation is difficult because of the collapse of the quantum state. Therefore, we need to find algorithms that use the laws of quantum mechanics in a very efficient and smart way so that this problem can be solved. Examples of such algorithms are the Shor’s and Grover’s algorithms. During a quantum computation process, the qubits evolve under a unitary evolution, that is, an evolution that can be represented by a unitary operator for which U † = U −1 . When such a computation is done, the computational system always interacts with the environment. Thus, to totally describe the system, we need a wavefunction that takes into account these interactions |ψiSE (system+environment). However, it is almost impossible to perfectly know the state of the environment, so we average over its possible states. This transforms pure state into mixed states that can be represented by the density matrix: X ρ= ωi |φi ihφi | (36) i

with the following properties: • ρ = ρ† • trρ = 1 • 0 ≤ ωi ≤ 1 Now, if the system and the environment evolve under a unitary operator and the environment is ignored, then we have a transformation called Complete-Positive map (CP-map). The CP-maps are the most general 13

transformation between physical quantities. In the case we are considering, the CP-map takes density matrices into other density matrices, that is, it preserves the properties listed above. As in classical computing, there is a set of universal quantum gates that can be used in a quantum computational problem. Examples of such gates are the Hadamar and the Control Phase gate. The 1-qubit Hadamar gate can be expressed as: ! 1 1 1 H=√ (37) 2 1 −1 and its action on the qubits is: 1 |0i → √ (|0i + |1i) 2

1 |1i → √ (|0i − |1i) . 2

An example of a 2-qubits Control Phase gate is:   1 0 0 0 0 1 0 0    B(φ) =   0 0 1 0  0 0 0 eiφ

(38)

(39)

Any n-qubits unitary evolution can be simulated by a combination of Hadamar and Control Phase gate.

7.3

Geometric Computation

The Berry’s phase allows us to do geometric computation. The advantage of this kind of computation is that it is more robust because the gates, which are holonomies, only depend on the areas in parameters space. Any error (like laser fluctuations or some other systematic errors) leave this areas invariant so the gates are not afected. An example of such geometrical computation is the two spin-1/2 particles system in a magnetic field. This section is similar to the one particle case described in section 6.2. Basically, we want to construct a gate that will give a geometric (Berry) phase to a two spins state. We start with the 2

14

particles kinetic Hamiltonian: ˆo = H =

=

ω1 ω2 σ z1 ⊗ 1 + 1 ⊗ σ z2 2 !2 ! ! ! ω1 1 0 ω2 1 0 1 0 1 0 ⊗ ⊗ + 2 0 −1 2 0 1 0 1 0 −1     1 0 0 0 1 0 0 0   ω1  0 0 1 0  ω2 0 −1 0 0   +   2 0 0 −1 0  2 0 0 1 0  0 0 0 −1 0 0 0 −1 

=

 ω1 + ω2 0 0 0  1 ω1 − ω2 0 0   0   2 0  0 −ω1 + ω2 0 0 0 0 −ω1 − ω2

(40)

Now, we add the following interaction term: ˆI H

= µσz1 ⊗ σz2 ! 1 0 = µ ⊗ 0 −1  1 0 0 0 −1 0  = µ 0 0 −1 0 0 0

! 1 0 0 −1  0 0   0 1

(41)

With the following definitions, ω− = ω1 − 2µ

ω+ = ω1 + 2µ

(42)

the total Hamiltonian can be written as: ˆ = H ˆo + H ˆI H   ω+ + ω2 0 0 0  1 0 ω− − ω2 0 0   =   2  0 0 −ω+ + ω2 0 0 0 0 −ω− − ω2 15

(43)

We see in this Hamiltonian that the energy eigenvalues corresponding to each state are as follow: | ↑↑i → ω+ + ω2

| ↑↓i → ω− − ω2

| ↓↑i → −ω+ + ω2

| ↓↓i → −ω− − ω2

(44)

Now, we want to extract a Berry’s phase out of these states by interacting only with the first spin. Thus, the evolution unitary operator will be of the form: U = U1 ⊗ 1 (45) with φ

θ

U1 = ei 2 σz1 ei 2 σy1

(46)

To achieve such an evolution, we interact with spin 1 with a magnetic field of frequency ν, amplitude α, phase φ and coupling constant λ. First, we consider the case for which the second spin is in the ”up” state. As in the example for a single spin (section 6.2), the Berry’s phase acquired by the state after such an evolution is given by: γ+ = π(1 − cos θ+ )

(47)

Here, cos θ+ is given by eq. 33 with the difference that the precession frequency of spin one is now given by ω+ = ω1 + 2µ. Thus, with the definition ∆+ = ω+ − ν, we can write: ∆+

cos θ+ = q

(48)

∆2+ + 4λ2 α2

and with the help of eq. 47, we get the desired Berry’s phase and the states are transformed in the following way: | ↓↑i → e−iγ+ | ↓↑i

| ↑↑i → eiγ+ | ↑↑i

(49)

Now, if the second qubit is in the ”down” state, the precession frequency of the first spin is equal to ω− = ω1 − 2µ and the Berry’s phase is still given by: γ− = π(1 − cos θ− ) (50) with

∆−

cos θ− = q

∆2− + 4λ2 α2

16

(51)

where we have used the definition ∆− = ω− − ν. The states transform in the following way: | ↓↓i → e−iγ− | ↓↓i

| ↑↓i → eiγ− | ↑↓i

(52)

However, this is not yet the gate we are looking for since, because of the time evolution of the system, the states also acquire a dynamical phase during the process. Thus, the correct transformation for the | ↑↑i state is given by: | ↑↑i → e−i(ω+ +ω2 )t eiγ+ | ↑↑i

(53)

where we have used the result of eq. 44. To eliminate the dynamical phase, we use the spin-echo technique which consists of a series of time evolutions between each we insert π − rotations of the spins in order to cancel out the dynamical phase. In the following, we will use the notation: • πi : π radian rotation of the ith spin • c: time evolution • ¯c : conjugate time evolution(give the complex conjugate of the Berry’s phase). Starting, for example, with the state | ↑↑i, we do the following series of transformations: π

c

→1 e−i(ω+ +ω2 )t eiγ+ | ↓↑i

c

→2 e−2iω2 t e2iγ+ | ↓↓i π →1 e2iγ+ −iγ− ei(ω− −ω2 )t | ↑↓i

c

π

| ↑↑i → e−i(ω+ +ω2 )t eiγ+ | ↑↑i → e−i(ω+ +ω2 )t e2iγ+ e−i(−ω+ +ω2 )t | ↓↑i c → e2iγ+ −iγ− e−2iω2 t e−i(−ω− −ω2 )t | ↓↓i

π

→ e2i(γ+ −γ− ) ei(ω− −ω2 )t e−i(ω− −ω2 )t | ↑↓i →2 e2i(γ+ −γ− ) | ↑↑i Thus, the result of this spin-echo technique is to give to the initial state a pure Berry’s phase, that is, this is the gate we were looking for: | ↑↑i → e2i∆γ | ↑↑i

(54)

with ∆γ = γ+ − γ− . For the other eigenstates, we get: | ↑↓i → e−2i∆γ | ↑↓i −2i∆γ

| ↓↑i → e

(55)

| ↓↑i

(56)

| ↓↓i → e2i∆γ | ↓↓i

(57)

17

This technique to obtain Berry’s phases works quite well for the one and two spin-1/2 cases (see above example). However, for more than two spins, the calculations become very tedious. So, the method to get Berry’s phases cannot be easily generalized to the n-spins case. Also, if the initial state of the spins is a superposition of eigenstates, for example |ψi = | ↑↑i + | ↑↓i, then the method described above cannot be applied to get Berry’s phases.

8

Adiabatic Holonomies

An adiabatic holonomy in quantum mechanics is a unitary transformation of geometric origin generated by slowly changing the Hamiltonian of a quantum system through a set of parameters.The parameters define a manifold called parameter space and the term geometric comes from the fact that the holonomies depend only on the path followed by the system in this space. Let us consider an Hamiltonian that depends on a parameters ~λ(t). Suppose further that the dependance of the Hamiltonian on ~λ is cyclic, that is, ∃ τ such that: ˆ ~λ(0)) = H( ˆ ~λ(τ )). H( (58) which means that during the time τ , ~λ has described a closed loop C in the parameter space. Let us consider the case where the energy level of the Hamiltonian are degenerate, that is, for a given eigenenergy Ek , there is more than one corresponding eigenstates: ˆ H(λ)|ψ n (λ)i = Ek (λ)|ψn (λ)i

(59)

where n ranges from 1 to the degree of degeneracy of the energy eigenvalue. When we slowly (with respect to any time scale associated with the system dynamics) evolve such a degenerate state, the adiabatic theorem tells us that the final state stays in the subspace defined by our energy level. Let us consider a degenerate subspace with energy eigenvalue Eo . We take our initial state to be |ψi0 i = |ψi (0)i, which belongs to the considered subspace. After a time τ (so that eq. 58 is applicable), this state is mapped to e−iEo τ UC (λ)|ψi0 i.The transformation UC (λ) is the holonomy that we want to determine. By a renormalization of the energy levels, we can take Eo = 0, which will simplify the following. In terms of the local reference basis of the degenerate subspace {|φα (λ)i}, the state at any point of the adiabatic path 18

in parameter space is expressed as: |ψα (λ)i = Uαβ (λ)|φβ (λ)i

(60)

Similarly for the bra state: † hψδ (λ)| = hφγ (λ)|Uγδ (λ)

(61)

Now, we substitute eq.60 into Schr¨odinger’s equation: ∂ |ψα (t)i ∂t 0 = 0 + U˙ αβ |φβ i + Uαβ |φ˙ β i 0 = U˙ αβ |φβ i + Uαβ |φ˙ β i

ˆ α (t)i = −iH|ψ

(62)

where the dot refer to the time derivative. We take the inner product of eq. 62 with eq. 61:   † hφγ |Uγδ U˙ αβ |φβ i + Uαβ |φ˙ β i = 0 † † ˙ hφγ |Uγδ Uαβ |φβ i + hφγ |Uγδ Uαβ |φ˙ β i = 0 † ˙ † Uγδ Uαβ Pβγ + Uγδ Uαβ Aβγ

= 0

(63)

where Pβγ = hφγ |φβ i and Aβγ = hφγ |φ˙ β i. P is an Hermitian matrix which is equal to the identity if the local basis vectors are orthogonal. Now, rearranging the terms: (U˙ P )αγ −1 ˙ I⇒ U U U −1 U˙

= −(U A)αγ −1 = −AP I = − AP −1 C H

C

⇒ UC

= Pˆ e−

C

AP −1

(64)

where Pˆ is the path ordering operator. In the last step. we have integrated over the closed path C in the parameters space. The dimensionality of the holonomy is equal to the degree of degeneracy of the subspace we are looking at. Berry phase is the special case when the eigenspace is non-degenerate so that the unitary transformation is then only a complex number.

19

9 9.1

Non-adiabatic Berry phase: A Generalization Cyclic Evolution

Now that we have considered the general case for holonomies in a degenerate eigenspace, let us consider again the non-degenerate geometric phase case. In sections 6 and 7, we computed Berry phases for systems whose initial states were one of their eigenstates. One can ask what happen when such systems start in a superposition of eigenstates such as |ψi = α| ↑i + β| ↓i. To answer this question, we will consider the phase change for a general cyclic evolution. Let H be an Hilbert space. Consider the normalized state |ψi ∈ H which evolves according to the Schr¨odinger’s equation: ∂ ˆ H|ψ(t)i = i |ψ(t)i ∂t

(65)

together with |ψ(τ )i = eiφ |ψ(0)i for a certain τ , φ real. This means that after a time τ , the quantum state comes back to the same ray of H in which it was initially. Thus |ψ(t)i describes a curve C : [0, τ ] → H. Let also Π : H → P be the projection map that project every rays of H onto a single point in P and is defined by:  Π(|ψi) = |ψ 0 i : |ψ 0 i = c|ψi, c ∈ C (66) Since all states in a ray of H are mapped under Π into a single point, the curve C in H is mapped to a closed curve Cˆ in P. An explicit form of Π can be defined by writing: ˜ |ψ(t)i = e−if (t) |ψ(t)i such that f (τ ) − f (0) = φ

(67)

˜ )i = |ψ(0)i. ˜ with |ψ(τ Thus, the ”tilde” states now make a cyclic evolution in their Hilbert space. Taking the time derivative of 67, we get: ∂ ˜ ∂ ˜ |ψ(t)i = if˙(t)|ψ(t)i + eif (t) |ψ(t)i ∂t ∂t

(68)

˜ Now, taking the inner product with the bra state hψ(t)| yield: ˜ hψ(t)|

∂ ∂ ˜ |ψ(t)i = if˙(t) + hψ(t)| |ψ(t)i ∂t ∂t ˙ ˆ = if (t) + hψ(t)|H|ψ(t)i 20

(69)

where we have used eq. 65 in closed curve C: I ∂ ˜ ˜ |ψ(t)i = hψ(t)| ∂t C Z τ ∂ ˜ ˜ hψ(t)| |ψ(t)idt = ∂t 0

the last step. Now we integrate along the I i ZCτ i

f˙ − i

I

ˆ hψ(t)|H|ψ(t)i Z τ ˆ f˙dt − i hψ(t)|H|ψ(t)idt C

0

0

Z τ ˆ = i (f (τ ) − f (0)) − i hψ(t)|H|ψ(t)idt 0 Z τ ˆ = iφ − i hψ(t)|H|ψ(t)idt (70) 0

Therefore, we can write: Z τ Z τ ∂ ˜ ˜ ˆ φ = −i hψ(t)| |ψ(t)idt + hψ(t)|H|ψ(t)idt ∂t 0 0

(71)

The second term of eq. 71 is recognized to be the usual dynamical phase that arises from the evolution of the state. Thus, let us define the geometrical phase to be: Z τ ∂ ˜ ˜ β = −i hψ(t)| |ψ(t)idt. (72) ∂t 0 This phase is a only a geometric property of the closed curve Cˆ in P, that is, ˆ and C. To see this, we remark that, by a clever choice it doesn’t depend on H ˜ ˆ of f (t), the same |ψ(t)i can be used for all curves C for which Π(C) = C. Since all information about the Hamiltonian is contained in C and that many different C’s can be mapped in the same Cˆ in P, β is independent of ˆ and depends only on the cyclic evolution described by C. ˆ H We see that equation 72 is very similar to the expression for the Berry phase derived above (eq. 14). However, they are many differences between these two. First, we did not use the adiabatic approximation in the derivaˆ Finally, as tion of 72. Secondly,|ψ(t)i is not necessary an eigenstate of H. described in the above paragraph, β is associated with a cyclic evolution (closed curve) in the projective Hilbert space P while Berry phase is associated with a closed loop in the parameter space of the Hamiltonian. In fact, we can show that it is neither necessary nor sufficient to go around a closed loop in parameter space to get a cyclic evolution in P. However, for a given cyclic evolution in P, one can find an Hamiltonian for which the 21

adiabatic approximation is valid and then compute the geometrical (Berry) phase with eq. 14 with the eigenstates of this Hamiltonian. Therefore, Berry phase, as described in the previous sections, in a special case of a more general situation involving geometrical phase.

9.2

An Example

Let us consider an example in which a spin-1/2 particle is placed in a magnetic field along the z-direction. It is initially in state |ψ(0)i = cos (θ/2)| ↑ ˆ It’s Hamiltonian is given i + sin (θ/2)| ↓i, which is not an eigenstate of H. by: ˆ = ω σz H (73) 2 At a time t later, the state of the spin is given by: ˆ

|ψ(t)i = e−iHt |ψ(0)i = e

iωt 2

cos (θ/2)| ↑i + e

−iωt 2

sin (θ/2)| ↓i

(74)

If we take ωt = 2π, then 

 2π |ψ t = i = −|ψ(0)i, ω

(75)

that is the state acquires an overall phase of π. Thus, under a projective map in P, this evolution is mapped into a closed curve. Therefore, this is a cyclic state. In order to calculate the geometric phase, let us first calculate the dynamical phase and then subtract it from the overall π phase. We have: Z 2π ω ˆ δ = hψ|H|ψidt 0

Z = 0

2π ω

 ω cos2 (θ/2) − sin2 (θ/2) dt 2

2π ω = cos (θ) ω 2 = π cos (θ)

(76)

Then, the geometrical phase is given by: β = π − δ = π(1 − cos (θ))

(77)

As we can see, this calculation does not require the adiabatic approximation nor the system to be in an eigenstate initially. 22