3, Coherent and Squeezed States 1. Coherent States 2. Squeezed States

Home , Coherent state, Displacement operator, Squeezed coherent state, Squeeze operator

3, Coherent and Squeezed States

1. Coherent states 2. Squeezed states 3. Field Correlation Functions 4. Hanbury Brown and Twiss experiment 5. Photon Antibunching 6. Quantum Phenomena in Simple Nonlinear Optics

Ref: Ch. 2, 4, 16 in ”Quantum Optics,” by M. Scully and M. Zubairy. Ch. 3, 4 in ”Mesoscopic Quantum Optics,” by Y. Yamamoto and A. Imamoglu. Ch. 6 in ”The Quantum Theory of Light,” by R. Loudon. Ch. 5, 7 in ”Introductory Quantum Optics,” by C. Gerry and P.Knight. Ch. 5, 8 in ”Quantum Optics,” by D. Wall and G. Milburn.

IPT5340, Fall ’06 – p. 1/50 Uncertainty relation

Non-commuting observable do not admit common eigenvectors.

Non-commuting observables can not have deﬁnite values simultaneously.

Simultaneous measurement of non-commuting observables to an arbitrary degree of accuracy is thus incompatible. variance: ∆Aˆ2 = Ψ (Aˆ Aˆ )2 Ψ = Ψ Aˆ2 Ψ Ψ Aˆ Ψ 2. h | −h i | i h | | i−h | | i 1 ∆A2∆B2 [ Fˆ 2 + Cˆ 2], ≥ 4 h i h i where [A,ˆ Bˆ]= iC,ˆ and Fˆ = AˆBˆ + BˆAˆ 2 Aˆ Bˆ . − h ih i Take the operators Aˆ =q ˆ (position) and Bˆ =p ˆ (momentum) for a free particle,

~2 [ˆq, pˆ]= i~ ∆ˆq2 ∆ˆp2 . →h ih i≥ 4

IPT5340, Fall ’06 – p. 2/50 Uncertainty relation if Re(λ) = 0, Aˆ + iλBˆ is a normal operator, which have orthonormal eigenstates. the variances,

iλ i ∆Aˆ2 = [ Fˆ + i Cˆ ], ∆Bˆ2 = [ Fˆ i Cˆ ], − 2 h i h i − 2λ h i− h i

set λ = λr + iλi,

2 1 2 1 2 ∆Aˆ = [λi Fˆ + λr Cˆ ], ∆Bˆ = ∆Aˆ , λi Cˆ λr Fˆ = 0. 2 h i h i λ 2 h i− h i | | if λ = 1, then ∆Aˆ2 =∆Bˆ2, equal variance minimum uncertainty states. | | if λ = 1 along with λ = 0, then ∆Aˆ2 =∆Bˆ2 and Fˆ = 0, uncorrelated equal | | i h i variance minimum uncertainty states. 2 λi 2 λ 2 1 if λr = 0, then Fˆ = Cˆ , ∆Aˆ = | | Cˆ , ∆Bˆ = Cˆ . 6 h i λr h i 2λr h i 2λr h i If Cˆ is a positive operator then the minimum uncertainty states exist only if λr > 0.

IPT5340, Fall ’06 – p. 3/50 Minimum Uncertainty State

(ˆq qˆ ) ψ = iλ(ˆp pˆ ) ψ −h i | i − −h i | i 2r if we deﬁne λ = e− , then

r r r r (e qˆ + ie− pˆ) ψ =(e qˆ + ie− pˆ ) ψ , | i h i h i | i the minimum uncertainty state is deﬁned as an eigenstate of a non-Hermitian operator erqˆ + ie rpˆ with a c-number eigenvalue er qˆ + ie r pˆ . − h i − h i the variances of qˆ and pˆ are

~ ~ 2 2r 2 2r ∆ˆq = e− , ∆ˆp = e . h i 2 h i 2 here r is referred as the squeezing parameter.

IPT5340, Fall ’06 – p. 4/50 Quantization of EM ﬁelds

ˆ ~ 1 the Hamiltonian for EM fields becomes: H = j ωj (âj†aˆj + 2 ), the electric and magnetic fields become, P

~ω ˆ j 1/2 iωj t iωj t Ex(z,t) = ( ) [ˆaj e− +a ˆj†e ] sin(kj z), ǫ0V Xj = cj [ˆa1j cos ωj t +a ˆ2j sin ωj t]uj (r), Xj

IPT5340, Fall ’06 – p. 5/50 Phase diagram for EM waves

Electromagnetic waves can be represented by

Eˆ(t)= E [Xˆ sin(ωt) Xˆ cos(ωt)] 0 1 − 2 where

Xˆ1 = amplitude quadrature

Xˆ2 = phase quadrature

IPT5340, Fall ’06 – p. 6/50 Quadrature operators the electric and magnetic ﬁelds become,

~ω ˆ j 1/2 iωj t iωj t Ex(z,t) = ( ) [âj e− +a ˆj†e ] sin(kj z), ǫ0V Xj = cj [â1j cos ωj t +a ˆ2j sin ωj t]uj (r), Xj note that aˆ and aˆ† are not hermitian operators, but (â†)† =a ˆ.

1 1 aˆ1 = (â +a ˆ ) and aˆ2 = (â aˆ ) are two Hermitian (quadrature) operators. 2 † 2i − † the commutation relation for aˆ and aˆ† is [â, aˆ†] = 1, the commutation relation for and is i , aˆ aˆ† [â1, aˆ2]= 2 and ∆â2 ∆â2 1 . h 1ih 2i≥ 16

IPT5340, Fall ’06 – p. 7/50 Minimum Uncertainty State

(â1 aˆ1 ) ψ = iλ(â2 aˆ2 ) ψ −h i | i − −h i | i 2r r r r r if we define λ = e , then (e aˆ1 + ie aˆ2) ψ =(e aˆ1 + ie aˆ2 ) ψ , − − | i h i − h i | i the minimum uncertainty state is defined as an eigenstate of a non-Hermitian operator eraˆ + ie raˆ with a c-number eigenvalue er aˆ + ie r aˆ . 1 − 2 h 1i − h 2i the variances of aˆ1 and aˆ2 are

2 1 2r 2 1 2r ∆ˆa = e− , ∆ˆa = e . h 1i 4 h 2i 4 here r is referred as the squeezing parameter. when r = 0, the two quadrature amplitudes have identical variances,

1 ∆ˆa2 = ∆ˆa2 = , h 1i h 2i 4

r r in this case, the non-Hermitian operator, e aˆ1 + ie− aˆ2 =a ˆ1 + iaˆ2 =a ˆ, and this minimum uncertainty state is termed a coherent state of the electromagnetic ﬁeld, an eigenstate of the annihilation operator, aˆ α = α α . | i | i

IPT5340, Fall ’06 – p. 8/50 Coherent States

aˆ α = α α . | i | i expand the coherent states in the basis of number states,

aˆn αn α = n n α = n 0 α = 0 α n , | i | ih | i | ih | √ | i √ h | i| i n n n! n n! X X X imposing the normalization condition, α α = 1, we obtain, h | i

m n 2 (α∗) α α 2 1= α α = m n = e| | 0 α , h | i h | i √ √ |h | i| n m m! n! X X we have 1 2 n α ∞ α α = e− 2 | | n , | i √n! | i nX=0

IPT5340, Fall ’06 – p. 9/50 Properties of Coherent States the coherent state can be expressed using the photon number eigenstates,

1 2 n α ∞ α α = e− 2 | | n , | i √n! | i nX=0 the probability of ﬁnding the photon number n for the coherent state obeys the Poisson distribution, 2 e α α 2n P (n) n α 2 = −| | | | , ≡ |h | i| n! the mean and variance of the photon number for the coherent state α are, | i

nˆ = nP (n)= α 2, h i | | n X ∆ˆn2 = nˆ2 nˆ 2 = α 2 = nˆ , h i h i−h i | | h i

IPT5340, Fall ’06 – p. 10/50 Poisson distribution

IPT5340, Fall ’06 – p. 11/50 Photon number statistics

For photons are independent of each other, the probability of occurrence of n photons, or photoelectrons in a time interval T is random. Divide T into N intervals, the probability to find one photon per interval is, p =n/N ¯ , the probability to find no photon per interval is, 1 p, − the probability to find n photons per interval is,

N! n N n P (n)= p (1 p) − , n!(N n)! − − which is a binomial distribution. when N , → ∞ n¯nexp( n¯) P (n)= − , n! this is the Poisson distribution and the characteristics of coherent light.

IPT5340, Fall ’06 – p. 12/50 Real life Poisson distribution

IPT5340, Fall ’06 – p. 13/50 Displacement operator coherent states are generated by translating the vacuum state 0 to have a ﬁnite | i excitation amplitude α,

1 2 n 1 2 n α ∞ α α ∞ (αaˆ†) α = e− 2 | | n = e− 2 | | 0 , | i √ | i n! | i n=0 n! n=0 1 2 X X α αaˆ† = e− 2 | | e 0 , | i

∗ since aˆ 0 = 0, we have e α aˆ 0 = 0 and | i − | i

1 2 α αaˆ† α∗aˆ α = e− 2 | | e e− 0 , | i | i any two noncommuting operators Aˆ and Bˆ satisfy the Baker-Hausdorff relation, 1 Aˆ+Bˆ Aˆ Bˆ [A,ˆ Bˆ] e = e e e− 2 , provided [A,ˆ [A,ˆ Bˆ]]=0, using Aˆ = αaˆ , Bˆ = α aˆ, and [A,ˆ Bˆ]= α 2, we have, † − ∗ | |

αaˆ† α∗aˆ α = Dˆ (α) 0 = e− − 0 , | i | i | i where Dˆ(α) is the displacement operator, which is physically realized by a classical oscillating current.

IPT5340, Fall ’06 – p. 14/50 Displacement operator the coherent state is the displaced form of the harmonic oscillator ground state,

αaˆ† α∗aˆ α = Dˆ (α) 0 = e− − 0 , | i | i | i where Dˆ(α) is the displacement operator, which is physically realized by a classical oscillating current, the displacement operator Dˆ(α) is a unitary operator, i.e.

1 Dˆ †(α)= Dˆ ( α)=[Dˆ (α)]− , −

Dˆ (α) acts as a displacement operator upon the amplitudes aˆ and aˆ†, i.e.

1 Dˆ − (α)ˆaDˆ (α) =a ˆ + α, 1 Dˆ − (α)ˆa†Dˆ (α) =a ˆ† + α∗,

IPT5340, Fall ’06 – p. 15/50 Radiation from a classical current the Hamiltonian (p A) that describes the interaction between the ﬁeld and the · current is given by V = J(r, t) Aˆ(r, t)d3r, · Z where J(r, t) is the classical current and Aˆ(r, t) is quantized vector potential,

1 iωkt+ik r Aˆ(r, t)= i Ekaˆke− · + H.c., − ωk Xk the interaction picture Schrödinger equation obeys,

d i Ψ(t) = V Ψ(t) , dt | i − ~ | i the solution is Ψ(t) = exp[α aˆ α aˆ ] 0 , where | i k k † − k∗ k | ik 1 t iωt′ ik r αk = Ek dt′ drJ(r, t)e − · , ~ωk 0 Q R R this state of radiation ﬁeld is called a coherent state,

α = (αaˆ† α∗aˆ) 0 . | i − | i

IPT5340, Fall ’06 – p. 16/50 Properties of Coherent States

the probability of ﬁnding n photons in α is given by a Poisson distribution, | i the coherent state is a minimum-uncertainty states, the set of all coherent states α is a complete set, | i 1 α α d2α = π n n , or α α d2α = 1, | ih | | ih | | ih | n π Z X Z two coherent states corresponding to different eigenstates α and β are not orthogonal,

1 2 1 2 1 2 α β = exp( α + α∗β β )= exp( α β ), h | i − 2 | | − 2 | | − 2 | − | coherent states are approximately orthogonal only in the limit of large separation of the two eigenvalues, α β , | − | → ∞

IPT5340, Fall ’06 – p. 17/50 Properties of Coherent States

therefore, any coherent state can be expanded using other coherent state,

1 2 1 2 1 2 β α α = d β β β α = d βe− 2 | − | β , | i π | ih | i π | i Z Z this means that a coherent state forms an overcomplete set, the simultaneous measurement of aˆ1 and aˆ2, represented by the projection operator α α , is not an exact measurement but instead an approximate | ih | measurement with a ﬁnite measurement error.

IPT5340, Fall ’06 – p. 18/50 q-representation of the coherent state

coherent state is deﬁned as the eigenstate of the annihilation operator,

aˆ α = α α , | i | i where aˆ = 1 (ωqˆ + ipˆ), √2~ω the q-representation of the coherent state is,

∂ (ωq + ~ ) q α = √2~ωα q α , ∂q h | i h | i with the solution,

ω ω p q α =( )1/4exp[ (q q )2 + i h i q + iθ], h | i π~ − 2~ −h i ~ where θ is an arbitrary real phase,

IPT5340, Fall ’06 – p. 19/50 Expectation value of the electric ﬁeld for a single mode electric ﬁeld, polarized in the x-direction,

Eˆx = E0[ˆa(t) +a ˆ†(t)] sin kz, the expectation value of the electric ﬁeld operator,

~ω ∆Eˆ(t)2 1/2 = sin kz , h i s 2ǫ0V | |

∆Eˆ(t)2 1/2 is independent of the ﬁeld strength α , h i | | quantum noise becomes less important as α 2 increases, or why a highly excited | | coherent state α 1 can be treated as a classical EM ﬁeld. | | ≫ IPT5340, Fall ’06 – p. 20/50 Phase diagram for coherent states

IPT5340, Fall ’06 – p. 21/50 Generation of Coherent States

In classical mechanics we can excite a SHO into motion by, e.g. stretching the spring to a new equilibrium position,

2 p 1 2 Hˆ = + kx eE0x, 2m 2 − p2 1 eE 1 eE = + k(x 0 )2 ( 0 )2, 2m 2 − k − 2 k

upon turning off the dc ﬁeld, i.e. E0 = 0, we will have a coherent state α which | i oscillates without changing its shape, applying the dc ﬁeld to the SHO is mathematically equivalent to applying the displacement operator to the state 0 . | i

IPT5340, Fall ’06 – p. 22/50 Generation of Coherent States a classical external force f(t) couples linearly to the generalized coordinate of the harmonic oscillator,

1 Hˆ = ~ω(âaˆ† + )+ ~[f(t)â + f ∗(t)â†], 2 for the initial state Ψ(0) = 0 , the solution is | i | i

Ψ(t) = exp[A(t)+ C(t)ˆa†] 0 , | i | i where

t t” t iω(t′ t”) iω(t′ t) A(t)= dt”f(t”) dt′e − f(t′), C(t)= i dt′e − f ∗(t′), − − Z0 Z0 Z0

When the classical driving force f(t) is resonant with the harmonic oscillator, iωt f(t)= f0e , we have

2 iωt 1 2 α C(t)= ie− f0t α, A(t)= (f0t) = | | , and Ψ(t) = α . − ≡ − 2 − 2 | i | i

IPT5340, Fall ’06 – p. 23/50 Attenuation of Coherent States

Glauber showed that a classical oscillating current in free space produces a multimode coherent state of light.

The quantum noise of a laser operating at far above threshold is close to that of a coherent state.

A coherent state does not change its quantum noise properties if it is attenuated, a beam splitter with inputs combined by a coherent state and a vacuum state 0 , | i

HˆI = ~κ(ˆa†ˆb +a ˆˆb†), interaction Hamiltonian where κ is a coupling constant between two modes, the output state is, with β = √T α and γ = √1 T α, −

Ψ out = Uˆ α a 0 = β a γ , with Uˆ = exp[iκ(ˆa†ˆb +a ˆˆb†)t], | i | i | ib | i | ib

The reservoirs consisting of ground state harmonic oscillators inject the vacuum ﬂuctuation and partially replace the original quantum noise of the coherent state.

Since the vacuum state is also a coherent state, the overall noise is unchanged.

IPT5340, Fall ’06 – p. 24/50 Coherent and Squeezed States

ˆ ˆ Uncertainty Principle: ∆X1∆X2 ≥ 1.

1. Coherent states: ∆Xˆ1 =∆Xˆ2 = 1,

2. Amplitude squeezed states: ∆Xˆ1 < 1,

3. Phase squeezed states: ∆Xˆ2 < 1, 4. Quadrature squeezed states.

IPT5340, Fall ’06 – p. 25/50 Squeezed States and SHO

Suppose we again apply a dc ﬁeld to SHO but with a wall which limits the SHO to a ﬁnite region, in such a case, it would be expected that the wave packet would be deformed or ’squeezed’ when it is pushed against the barrier.

Similarly the quadratic displacement potential would be expected to produce a squeezed wave packet,

2 p 1 2 2 Hˆ = + kx eE0(ax bx ), 2m 2 − − where the ax term will displace the oscillator and the bx2 is added in order to give us a barrier, 2 p 1 2 Hˆ = + (k + 2ebE0)x eaE0x, 2m 2 −

We again have a displaced ground state, but with the larger effective spring constant k′ = k + 2ebE0.

IPT5340, Fall ’06 – p. 26/50 Squeezed Operator

To generate squeezed state, we need quadratic terms in x, i.e. terms of the form 2 (ˆa +a ˆ†) , for the degenerate parametric process, i.e. two-photon, its Hamiltonian is

2 2 Hˆ = i~(gaˆ† g∗aˆ ), − where g is a coupling constant. the state of the ﬁeld generated by this Hamiltonian is

2 2 Ψ(t) = exp[(gaˆ† g∗aˆ )t] 0 , | i − | i deﬁne the unitary squeeze operator

1 2 1 2 Sˆ(ξ)= exp[ ξ∗aˆ ξaˆ† ] 2 − 2 where ξ = rexp(iθ) is an arbitrary complex number.

IPT5340, Fall ’06 – p. 27/50 Properties of Squeezed Operator deﬁne the unitary squeeze operator

1 2 1 2 Sˆ(ξ)= exp[ ξ∗aˆ ξaˆ† ] 2 − 2 where ξ = rexp(iθ) is an arbitrary complex number. squeeze operator is unitary, Sˆ (ξ)= Sˆ 1(ξ)= Sˆ( ξ) ,and the unitary † − − transformation of the squeeze operator,

iθ Sˆ†(ξ)ˆaSˆ(ξ) =a ˆ cosh r aˆ†e sinh r, − iθ Sˆ†(ξ)ˆa†Sˆ(ξ) =a ˆ† cosh r aeˆ − sinh r, −

Aˆ ˆ Aˆ ˆ ˆ ˆ 1 ˆ ˆ ˆ with the formula e Be− = B +[A, B]+ 2! [A, [A, B]],...

A squeezed coherent state α, ξ is obtained by ﬁrst acting with the displacement | i operator Dˆ (α) on the vacuum followed by the squeezed operator Sˆ(ξ), i.e.

α, ξ = Sˆ(ξ)Dˆ (α) 0 , | i | i with α = α exp(iψ). | |

IPT5340, Fall ’06 – p. 28/50 Uncertainty relation if Re(λ) = 0, Aˆ + iλBˆ is a normal operator, which have orthonormal eigenstates. the variances,

iλ i ∆Aˆ2 = [ Fˆ + i Cˆ ], ∆Bˆ2 = [ Fˆ i Cˆ ], − 2 h i h i − 2λ h i− h i

set λ = λr + iλi,

IPT5340, Fall ’06 – p. 29/50 Minimum Uncertainty State

(â1 aˆ1 ) ψ = iλ(â2 aˆ2 ) ψ −h i | i − −h i | i 2r if we define λ = e− , then

r r r r (e aˆ + ie− aˆ ) ψ =(e aˆ + ie− aˆ ) ψ , 1 2 | i h 1i h 2i | i the minimum uncertainty state is deﬁned as an eigenstate of a non-Hermitian r r r r operator e aˆ1 + ie aˆ2 with a c-number eigenvalue e aˆ1 + ie aˆ2 . − h i − h i the variances of aˆ1 and aˆ2 are

2 1 2r 2 1 2r ∆ˆa = e− , ∆ˆa = e . h 1i 4 h 2i 4

IPT5340, Fall ’06 – p. 30/50 Squeezed State deﬁne the squeezed state as

Ψs = Sˆ(ξ) Ψ , | i | i where the unitary squeeze operator

iθ Sˆ†(ξ)ˆaSˆ(ξ) =a ˆ cosh r aˆ†e sinh r, − iθ Sˆ†(ξ)ˆa†Sˆ(ξ) =a ˆ† cosh r aeˆ − sinh r, − for Ψ is the vacuum state 0 , the Ψs state is the squeezed vacuum, | i | i | i

ξ = Sˆ(ξ) 0 , | i | i

IPT5340, Fall ’06 – p. 31/50 Squeezed Vacuum State

for Ψ is the vacuum state 0 , the Ψs state is the squeezed vacuum, | i | i | i

ξ = Sˆ(ξ) 0 , | i | i the variances for squeezed vacuum are

1 ∆ˆa2 = [cosh2 r + sinh2 r 2 sinh r cosh r cos θ], 1 4 − 1 ∆ˆa2 = [cosh2 r + sinh2 r + 2 sinh r cosh r cos θ], 2 4 for θ = 0, we have

2 1 2r 2 1 +2r ∆ˆa = e− , and ∆ˆa = e , 1 4 2 4 and squeezing exists in the aˆ1 quadrature. for θ = π, the squeezing will appear in the aˆ2 quadrature.

IPT5340, Fall ’06 – p. 32/50 Quadrature Operators deﬁne a rotated complex amplitude at an angle θ/2

iθ/2 iθ/2 Yˆ1 + iYˆ2 = (ˆa1 + iaˆ2)e− =ae ˆ − , where

Yˆ1 cos θ/2 sin θ/2 aˆ1 = 0 Yˆ2 1 0 sin θ/2 cos θ/2 1 0 aˆ2 1 − @ A @ A @ A

r r then Sˆ†(ξ)(Yˆ1 + iYˆ2)Sˆ(ξ)= Yˆ1e− + iYˆ2e , the quadrature variance

2 1 2r 2 1 +2r 1 ∆Yˆ = e− , ∆Yˆ = e , and ∆Yˆ1∆Yˆ2 = , 1 4 2 4 4 in the complex amplitude plane the coherent state error circle is squeezed into an error ellipse of the same area, the degree of squeezing is determined by r = ξ which is called the squeezed | | parameter.

IPT5340, Fall ’06 – p. 33/50 Vacuum, Coherent, and Squeezed states

vacuum coherent squeezed-vacuum

amp-squeezed phase-squeezed quad-squeezed

IPT5340, Fall ’06 – p. 34/50 Squeezed Coherent State

A squeezed coherent state α, ξ is obtained by ﬁrst acting with the displacement | i operator Dˆ (α) on the vacuum followed by the squeezed operator Sˆ(ξ), i.e.

α, ξ = Dˆ(α)Sˆ(ξ) 0 , | i | i where Sˆ(ξ)= exp[ 1 ξ aˆ2 1 ξaˆ 2], 2 ∗ − 2 † for ξ = 0, we obtain just a coherent state. the expectation values,

2 2 iθ 2 2 α, ξ aˆ α, ξ = α, aˆ = α e sinh r cosh r, and aˆ†aˆ = α + sinh r, h | | i h i − h i | | with helps of Dˆ †(α)ˆaDˆ(α) =a ˆ + α and Dˆ †(α)ˆa†Dˆ(α) =a ˆ† + α∗, for r 0 we have coherent state, and α 0 we have squeezed vacuum. → → furthermore

iθ/2 2 1 2r 2 1 +2r α, ξ Yˆ1 + iYˆ2 α, ξ = αe− , ∆Yˆ = e− , and ∆Yˆ = e , h | | i h 1 i 4 h 2 i 4

IPT5340, Fall ’06 – p. 35/50 Squeezed State from the vacuum state aˆ 0 = 0, we have | i

Sˆ(ξ)âSˆ†(ξ)Sˆ(ξ) 0 = 0, or Sˆ(ξ)âSˆ†(ξ) ξ = 0, | i | i since Sˆ(ξ)âSˆ (ξ) =a ˆ cosh r +a ˆ eiθ sinh r µaˆ + νaˆ , we have, † † ≡ †

(µaˆ + νaˆ†) ξ = 0, | i the squeezed vacuum state is an eigenstate of the operator µaˆ + νaˆ† with eigenvalue zero. similarly,

Dˆ(α)Sˆ(ξ)ˆaSˆ†(ξ)Dˆ †(α)Dˆ(α) ξ = 0, | i with the relation Dˆ (α)ˆaDˆ (α) =a ˆ α, we have † −

(µaˆ + νaˆ†) α, ξ =(α cosh r + α∗ sinh r) α, ξ γ α, ξ , | i | i≡ | i

IPT5340, Fall ’06 – p. 36/50 Squeezed State and Minimum Uncertainty State

write the eigenvalue problem for the squeezed state

(µaˆ + νaˆ†) α, ξ =(α cosh r + α∗ sinh r) α, ξ γ α, ξ , | i | i≡ | i

iθ/2 in terms of in terms of aˆ =(Yˆ1 + iYˆ2)e we have

2r (Yˆ1 + ie− Yˆ2) α, ξ = β1 α, ξ , | i | i

where r iθ/2 2r β = γe− e− = Yˆ + i Yˆ e− , 1 h 1i h 2i

in terms of aˆ1 and aˆ2 we have

(ˆa1 + iλaˆ†) α, ξ = β2 α, ξ , 2 | i | i

where µ ν γ λ = − , and β2 = , µ + ν µ + ν

IPT5340, Fall ’06 – p. 37/50 Squeezed State in the basis of Number states consider squeezed vacuum state ﬁrst,

∞ ξ = Cn n , | i | i nX=0 with the operator of (µaˆ + νaˆ ) ξ = 0, we have † | i

ν n 1/2 Cn+1 = ( ) Cn 1, − µ n + 1 − only the even photon states have the solutions,

m iθ m (2m 1)!! 1/2 C2m =( 1) (e tanh r) [ − ] C0, − (2m)!! where C0 can be determined from the normalization, i.e. C0 = √cosh r, the squeezed vacuum state is

1 ∞ (2m)! ξ = ( 1)m eimθ tanhm r 2m , | i √ − 2mm! | i cosh r m=0 p X

IPT5340, Fall ’06 – p. 38/50 Squeezed State in the basis of Number states the squeezed vacuum state is

1 ∞ (2m)! ξ = ( 1)m eimθ tanhm r 2m , | i √ − 2mm! | i cosh r m=0 p X the probability of detecting 2m photons in the ﬁeld is

2m 2 (2m)! tanh r P2m = 2m ξ = , |h | i| 22m(m!)2 cosh r

for detecting 2m + 1 states P2m+1 = 0, the photon probability distribution for a squeezed vacuum state is oscillatory, vanishing for all odd photon numbers, the shape of the squeezed vacuum state resembles that of thermal radiation.

IPT5340, Fall ’06 – p. 39/50 Number distribution of the Squeezed State

0.8

0.6 n P

0.4

0.2

0 0 2 4 6 8 10 n

IPT5340, Fall ’06 – p. 40/50 Number distribution of the Squeezed Coherent State

For a squeezed coherent state,

1 n 2 ( 2 tanh r) 2 1 2 iθ 2 iθ 2 iθ 1/ Pn = n α, ξ = exp[ α (α∗ e +α e− ) tanh r]H (γ(e sinh(2r))− |h | i| n! cosh r −| | − 2 n

0.1

0.08

0.06

0.04

0.02

Ref: 30 40 50 60 70 80 Ch. 5, 7 in ”Introductory Quantum Optics,” by C. Gerry and P.Knight.

IPT5340, Fall ’06 – p. 41/50 Number distribution of the Squeezed Coherent State

A squeezed coherent state α, ξ is obtained by ﬁrst acting with the displacement | i operator Dˆ (α) on the vacuum followed by the squeezed operator Sˆ(ξ), i.e.

α, ξ = Dˆ(α)Sˆ(ξ) 0 , | i | i

the expectation values, 2 2 aˆ†aˆ = α + sinh r, h i | |

0.08 0.012

0.01 0.06 0.008

0.04 0.006

0.004 0.02 0.002

50 100 150 200 50 100 150 200

α 2 = 50, θ = 0, r = 0.5 α 2 = 50, θ = 0, r = 4.0 | | | |

IPT5340, Fall ’06 – p. 42/50 Generations of Squeezed States

Generation of quadrature squeezed light are based on some sort of parametric process utilizing various types of nonlinear optical devices. for degenerate parametric down-conversion, the nonlinear medium is pumped by a

ﬁeld of frequency ωp and that ﬁeld are converted into pairs of identical photons, of frequency ω = ωp/2 each,

(2) 2 2 Hˆ = ~ωaˆ†aˆ + ~ωpˆb†ˆb + i~χ (ˆa ˆb† aˆ† ˆb), − where b is the pump mode and a is the signal mode. assume that the ﬁeld is in a coherent state βe iωpt and approximate the | − i iωpt iωpt operators ˆb and ˆb† by classical amplitude βe− and β∗e , respectively, we have the interaction Hamiltonian for degenerate parametric down-conversion,

2 2 Hˆ = i~(η∗aˆ ηaˆ† ), I − where η = χ(2)β.

IPT5340, Fall ’06 – p. 43/50 Generations of Squeezed States we have the interaction Hamiltonian for degenerate parametric down-conversion,

2 2 Hˆ = i~(η∗aˆ ηaˆ† ), I − where η = χ(2)β, and the associated evolution operator,

2 2 Uˆ (t)= exp[ iHˆ t/¯]= exp[(η∗aˆ ηaˆ† )t] Sˆ(ξ), I − I − ≡ with ξ = 2ηt. for degenerate four-wave mixing, in which two pump photons are converted into two signal photons of the same frequency,

(3) 2 2 2 2 Hˆ = ~ωaˆ†aˆ + ~ωˆb†ˆb + i~χ (ˆa ˆb† aˆ† ˆb ), − the associated evolution operator,

2 2 Uˆ (t)= exp[(η∗aˆ ηaˆ† )t] Sˆ(ξ), I − ≡ with ξ = 2χ(3)β2t.

IPT5340, Fall ’06 – p. 44/50 Generations of Squeezed States

Nonlinear optics:

Courtesy of P. K. Lam

IPT5340, Fall ’06 – p. 45/50 Generation and Detection of Squeezed Vacuum

1. Balanced Sagnac Loop (to cancel the mean ﬁeld),

2. Homodyne Detection.

M. Rosenbluh and R. M. Shelby, Phys. Rev. Lett. 66, 153(1991).

IPT5340, Fall ’06 – p. 46/50 Beam Splitters

Wrong picture of beam splitters,

aˆ2 = raˆ1, aˆ3 = taˆ1, where r and t are the complex reﬂectance and transmittance respectively which require that r 2 + t 2 = 1. | | | | in this case,

2 2 2 2 [â2, aˆ†]= r [â2, aˆ†]= r , [â3, aˆ†]= t [â2, aˆ†]= t , and [â2, aˆ†]= rt∗ = 0, 2 | | 2 | | 3 | | 2 | | 3 6 this kind of the transformations do not preserve the commutation relations.

Correct transformations of beam splitters,

aˆ2 r jt aˆ0 = , 0 aˆ3 1 0 jt r 1 0 aˆ1 1 @ A @ A @ A

IPT5340, Fall ’06 – p. 47/50 Homodyne detection

the detectors measure the intensities Ic = cˆ cˆ and I = dˆ dˆ , and the h † i d h † i difference in these intensities is,

Ic I = nˆ = cˆ†cˆ dˆ†dˆ = i aˆ†ˆb aˆˆb† , − d h cdi h − i h − i assuming the b mode to be in the coherent state βe iωt , where β = β e iψ, we | − i | | − have iωt iθ iωt iθ nˆ = β aeˆ e− +a ˆ†e− e , h cdi | |{ } where θ = ψ + π/2, assume that a mode light is also of frequency ω (in practice both the a and b iωt modes derive from the same laser), i.e. aˆ =a ˆ0e− , we have

nˆ = 2 β Xˆ(θ) , h cdi | |h i ˆ 1 iθ iθ where X(θ)= 2 (â0e− +a ˆ0†e ) is the field quadrature operator at the angle θ, by changing the phase ψ of the local oscillator, we can measure an arbitrary quadrature of the signal field.

IPT5340, Fall ’06 – p. 48/50 Detection of Squeezed States mode a contains the single field that is possibly squeezed, mode b contains a strong coherent classical field, local oscillator, which may be taken as coherent state of amplitude β, for a balanced homodyne detection, 50 : 50 beam splitter, the relation between input (â, ˆb) and output (ˆc, dˆ) is,

1 1 cˆ = (ˆa + iˆb), dˆ= (ˆb + iaˆ), √2 √2

the detectors measure the intensities Ic = cˆ cˆ and I = dˆ dˆ , and the h † i d h † i difference in these intensities is,

Ic I = nˆ = cˆ†cˆ dˆ†dˆ = i aˆ†ˆb aˆˆb† , − d h cdi h − i h − i

IPT5340, Fall ’06 – p. 49/50 Squeezed States in Quantum Optics

Generation of squeezed states: nonlinear optics: χ(2) or χ(3) processes, cavity-QED, photon-atom interaction, photonic crystals, semiconductor, photon-electron/exciton/polariton interaction,

··· Applications of squeezed states: Gravitational Waves Detection, Quantum Non-Demolition Measurement (QND), Super-Resolved Images (Quantum Images), Generation of EPR Pairs, Quantum Informatio Processing, teleportation, cryptography, computing,

···

IPT5340, Fall ’06 – p. 50/50