Positive P-Representations of the Thermal Operator from Quantum Control Theory

0.15 MB·English

by  John A. Sidles

Positive P-Representations of the Thermal Operator
from Quantum Control Theory
John A. Sidles∗
Quantum System Engineering Group
University of Washington, School of Medicine
Box 356500, Seattle, WA 98195, USA
(Dated: January 26, 2004)
A positive P-representation for the spin-j thermal density matrix is given in closed form. The
representation is constructed by regarding the wave function as the internal state of a closed-loop
control system. A continuous interferometric measurement process is proved to einselect coherent
states, and feedback control is proved to be equivalent to a thermal reservoir. Ito equations are
derived, and the P-representation is obtained from a Fokker-Planck equation. Langevin equations
4 are derived, and the force noise is shown to be the Hilbert transform of the measurement noise.
0
Theformalism isappliedtomagneticresonanceforcemicroscopy(MRFM)andgravitywave(GW)
0
interferometry. Someunsolvedproblemsrelating todrift and diffusion on Hilbertspaces arenoted.
2

n PACSnumbers: 42.50.Lc,02.30.Yy,03.65.Yz,05.40.-a
a
J
7 Givenspin-j operatorss=(s1,s2,s3),itiswellknown Our main result is:
2 [1,2,3,8]thatacoherentstate xˆ canbeassociatedwith
each unit vector xˆ, such that |xˆixˆ = 1, xˆ sxˆ = jxˆ, Theorem 1. A positive P-sequence for the thermal op-
1 and a resolution of the identityho|peirator Ihis| | i erator is Pj(xˆ)=cj/Qj+1(−xˆ).
v
5 I= 2j+1 d2xˆ xˆ xˆ . (1) The normalization coefficient cj is readily found from
6 4π | ih | (2a) by taking the trace, with the result
1 Z4π
1 Given an operator ρj on each member of a sequence 2sinh((j+ 1)β)
040 aosefQqsup-esinenqc-uejeHtnocilebbeetoratbssepetaacosefestfjuonf∈cftu{ino0cn,ts12io,{1nP,sj.(.xQˆ.}),}(w,xˆae)nd,desfisaintmiesifalyairPnlyg- cj = Q−j+11/2(tˆ)−Q−j+121/2(−tˆ). (6a)
h/ { j } We first prove Theorem 1 nonconstructively by treat-
ingitasanansatz. Wetakethe thermalaxistˆ=(0,0,1)
p
t- ρj = 2j+1 d2xˆPj(xˆ) xˆ xˆ , (2a) such that hj,m|ρth|j,m′i = δmm′e−βm. Taking matrix
n 4π Z4π | ih | elements of (2a), Theorem 1 is proved if Pj satisfies
a xˆ ρ xˆ =Q (xˆ). (2b)
u h | j| i j 2j+1
q The properties of P-sequences and Q-sequences will be e−βmδmm′ = 4π d2xˆPj(xˆ)hj,m|xˆihxˆ|j,m′i. (7)
: Z4π
v the main topic of this article.
i Q-sequencespose few mathematicalmysteriesbecause We write xˆ =D(φ,θ,0)j,j , where the rotation
X | i | i
(2b) suffices to evaluate Q (xˆ) directly. In contrast, lit-
r j D(φ,θ,ψ)=e−iφs3e−iθs2e−ψs3 (8)
a tleisknownaboutPj-representations,particularlyabout
their positivity properties. The main obstruction is that
carries tˆ=(0,0,1) into xˆ =(sinθcosφ,sinθsinφ,cosθ).
P is nonunique due to the identity [1, 2]
j By an identity due to Wigner [4, 5]
d2xˆ Yl(xˆ) xˆ xˆ =0 for l >2j, (3)
m | ih | j,mxˆ = j,mD(φ,θ,0)j,j =Dj (φ,θ,0)
Z4π h | i h | | i mj
whereYl(xˆ)isasphericalharmonic. Tocircumventthis = 2j 1/2e−imφcosj+m1θ sinj−m1θ, (9)
m j−m 2 2
P-representation ambiguity we will apply methods from
(cid:0) (cid:1)
quantum measurement and control theory. we transform (7) into an integral representation of the
We willstudy the P-sequenceofthe thermaloperator. hypergeometric function F . Then via the identity [6]
2 1
F (2+2j,1+j+m,2+2j,z)=(1 z)−(j+m+1). (10)
ρth =exp( βtˆ·s), (4) 2 1 −
−
Theorem 1 follows immediately.
where tˆis the thermal axis. The Q-sequence of ρth can
Butwheredidtheansatz comefrom? Whydopositive
be computed by direct evaluation of (2b), and the well-
P-representations even exist for thermal operators?
known result [1, 3] is
We will now give a constructive proof that answers
Q (xˆ)=(cosh1β xˆ·tˆsinh1β)2j. (5) these questions. Our notation and methods are adapted
j 2 − 2
2
from Handbook of Stochastic Processes [7]. Purely alge- Here we have used einselect in Zurek’s sense [9]:
braicdetails arenotgiven,but whennon-obviousidioms
Einselection [excludes] all but a small set
or strategies are employed, we will show them.
[of states] from within a much larger Hilbert
Theproofconsistsofthreephysically-motivatedsteps.
space. Einselected states are distinguished by
In the first step, we will regard ψ as the internal state
| i their stability in spite of the monitoring envi-
of an open-loop dynamical system, and we will specify a
ronment.
sensor that monitors the spin axis.
In the second step, we will install a controller that Now we operate three spinometers simultaneously, with
uses the (imperfect) sensor measurements to align the generators (s1,s2,s3). We call this an open-loop triaxial
spinaxis withthe thermalaxis t. We willprovethatthe spinometer. We define a spin covariance σn
control noise is precisely equivalent to a thermal bath. (σ ) = ψ s s ψ ψ s ψ ψ s ψ (17)
n kl n k l n n k n n l n
In the third and final step, we will construct Ito and h | | i−h | | ih | | i
Fokker-Planck equations for the observed states. Theo- andsimilar to (13) and(15) we define an n’th stochastic
rem 1 then emerges constructively, with P as the solu- increment for the i′th spinometer
j
tion of a Fokker-Planck equation.
δE [σ]=E[σ ] E[σ ]. (18)
To begin Step 1, we define an open-loop uniaxial spin- i,n n+1 |s=si − n
ometer withgenerator stobetheMarkovchainofquan- Bystraightforwardstochasticanalysissimilarto(16),the
tum states (see, e.g., [8]) defined by net covariance increment is calculated to be
3
|ψn+1i=(AB||ψψnnii//√√PPAB;; PPBA ==hhψψnn||BA††AB||ψψnnii (11) trδEn[σ]=Xi=1trδEi,n[σ]=−4θ2trE[σn·σn⋆]. (19)
Physical intuition then suggests that similar to (14)
with increment operators A and B given by
=0 if and only if ψ is coherent,
A=[cos(θs)+sin(θs)]/√2, (12a) trσ ·σ⋆ | ni (20)
n n(>0 otherwise.
B =[cos(θs) sin(θs)]/√2. (12b)
− To prove this we define a spin vector x = ψ sψ /j,
h | | i
The generator s can be any Hermitian operator. such that σ σ = iǫ jx . Without loss of gen-
lm ml lmn n
Here P +P = 1 is guaranteed by A†A+B†B = I. erality we choo−se a frame in which x = (0,0,x ). We
A B 3
The measurementstrength is governedby θ, and we will define the symmetric tensorσ¯ = 1(σ +σ ), andwe
lm 2 lm ml
assume θ 1. Terms of order θ3 and higher will turn temporarily regard x and σ¯ ;k l as a set of seven
≪ 3 { kl ≥ }
out to be nonleading and we will ignore them. arbitrary real numbers. Then solely by reordering terms
We define the operator variance ∆(s) to be we construct two algebraic equalities
∆n(s)= ψn s2 ψn ψn sψn 2. (13) trσ·σ⋆ =pa+2pb+21pc+jpd+21p2d+12pe+12pf (21a)
h | | i−h | | i
trσ¯·σ¯ = 1j2+p +2p +1p +jp +1p2+1p (21b)
It is readily shown that in general 2 a b 2 c d 2 d 2 f
whose individual terms are
=0 if ψ is an eigenstate of s,
∆ (s) | ni (14) p =(σ¯ x2)2 p =j2 σ¯
n (>0 otherwise. a 33− 3 e − 33
p =σ¯2 +σ¯2 +σ¯2 p =j2(1 x2)
b 12 13 23 f − 3
For E an ensemble average,the variance increment p =(σ¯ σ¯ )2
c 11 22
−
δEn[∆(s)]=E[∆n+1(s)]−E[∆n(s)] (15) pd =(trσ¯ −σ¯33)2−(j(j+1)−σ¯33)2.
Byconstruction,eachtermisnonnegativeforall ψ and
is found by straightforwardstochastic analysis to be | i
zero for coherent ψ . In particular, p and p vanish if
e f
| i
δEn[∆(s)]=−4θ2E[∆2n(s)]. (16) anjd, io.en.l,yififaσn¯3d3o=nlyhψif|s23ψ|ψiis=cojh2eraenndt.jTxh3e=n (h2ψ0|)s3fo|ψlloiw=s
± | i
The sequence E[∆ (s)],E[∆ (s)],... is positive by immediately from the nonnegativity of (21a). Similarly
1 2
{ }
(14) and decreasing by (16), therefore it has a limit. (21b) implies a related inequality for σ¯:
This implies a vanishing increment (16), which implies
limn→∞En[∆2(s)] = 0, which implies limn→∞∆n(s) = trσ¯·σ¯ = 12j2 if and only if |ψi is coherent, (22)
0 for every Markov chain in the ensemble (except a set (> 1j2 otherwise.
2
of measure zero). This proves
Then by the same reasoning as for Theorem 2, (19)
Theorem 2. Open-loop uniaxial spinometers asymptot- implies that lim trσ ·σ⋆ =0 for every chain in the
n→∞ n n
ically einselect eigenstates of the generator. ensemble (except a set of measure zero). This proves
3
Theorem 3. Open-loop triaxial spinometers asymptoti- Now we turn our attention to Step 3, and focus onIto
cally einselect coherent states. and Fokker-Planck equations.
The following idioms lead quickly to Theorem 1. We
This completes Step 1 of our program. define a data three-vector d =(d ,d ,d ) by
n 1,n 2,n 3,n
Remark: itseemsreasonablethatall Liegroupsmight
asymptotically einselect coherent states [2], but the au- +1 for ψ Ac ψ ,
d = | n+1i∝ i| ni
thoNrohwaswsetubdeigeidnoSntelypS2U, a(2n)dPfo-rceupsreosnenctoanttiroonls.and ther- i,n (−1 for |ψn+1i∝Bic|ψni.
modynamics. For tˆthe thermal axis defined in (4), we Thus d ,d ,... is binary-valued data. The mean
1 2
{ }
modify the spinometer matrices (12b) such that
E[d ]= ψ Ac†Ac ψ ψ Bc†Bc ψ (27)
n h n| k k| ni−h n| k k| ni
Ac =e−iα(tˆ×s)k[cos(θs )+sin(θs )]/√2, (23a)
k k k satisfies
Bc =e+iα(tˆ×s)k[cos(θs ) sin(θs )]/√2. (23b)
k k − k E[dn]=2θE[xn] gsE[xn] (28)
≡
We will call this a closed-loop triaxial spinometer with
whichdefinesg =2θasthesensorgain. Weremarkthat
unitary feedback, because the operators exp( iαtˆ s) d /g is therefosre an unbiased measure of x .
act cumulatively to align the spin axis with tˆ.± × nWesnextdefine azero-meanstochasticvarinableW by
n
Closing the control loop does not alter the coherent
einselection because the sole effect of a post hoc unitary
operator on σn is a spatial rotation. Since trσn·σn⋆ is a dn =gsxn+Wn, (29)
scalar, (19) still holds. Thus we have
suchthat(toleadingorderinθ)W hasthesecond-order
n
Lemma1. Closed-looptriaxialspinometerswithunitary stochastic properties of a discrete Wiener increment:
feedback asymptotically einselect coherent states. E[(Wn)k(Wn′)k′]=δnn′δkk′. (30)
An ensemble of closed-loop spinometers has a density
An identity valid for ψ a coherent state,
n
matrix sequence ρ ,ρ ,... whose increment is | i
1 2
{ } ψ s s ψ = 1jδ +j(j 1)x x + 1iǫ x , (31)
3 h n| k l| ni 2 kl − 2 k j 2 klm m
δρ = Ac†ρ Ac +Bc†ρ Bc ρ , (24) gives rise to an Ito increment of conventional form
n k n k k n k− n
Xk=1(cid:16) (cid:17) δx =x x =g2a(x )+g b(x )·W , (32)
n n+1− n s n s n n
and we readily compute that δρ =0 for ρ =ρth, with
n n where a is called the drift vector and b is called the dif-
ρth the thermal operator defined in (4), provided
fusion matrix.
α= tanh1β or 1/α= tanh1β. (25) As a check, Lemma 1 requires that the stochastic mo-
− 4 − 4 tion of x be confined to a unit sphere, and so does the
n
We will show later on that ρth solves δρ = 0 uniquely, coherent state identity (31), since it is inhomogenous in
n x. For consistency, therefore, the increment of the m’th
because the Fokker-Planck equation for ρ has a unique
radialmoment must vanish for all m when x =1. This
stationary solution. This proves
| |
increment is readily calculated have the general form
Theorem 4. The density matrix of an ensemble of
closed-loop triaxial spinometers with unitary feedback is δEn[|x|m]∝ 21m(m−2)[xn·b(xn)·b†(xn)·xn],
asymptotically thermal. +mx 2[1trb(x ) b†(x )+x ·a(x )]. (33)
| n| 2 n · n n n
To connect (24) with the thermodynamic literature, Upon computing a(x) and b(x) explicitly we find
we set tˆ = (0,0,1) and expand to order θ2. The re-
sultisequivalenttoathermalmodelgivenbyPerelomov a(x)= 1x[α(1 2j)x·tˆ (1+ 1α2)]
4 − − 2
(pereq.23.2.1of[1]). Gardinergivessimilarmodel,(per + 1tˆ[α(1+2j) 1α2x·tˆ], (34a)
eq. 10.4.2 of [7]). In Lindblad form we find 4 − 2
b(x)= 1[I x x+α(tˆ x x·tˆI)], (34b)
2 − ⊗ ⊗ −
δρ = 1γ(ν+1)(s s ρ 2s ρs +s s )
n −2 + − − − + + − such that the radial increment (33) indeed vanishes.
−+12γθν2((ss3−ss3+ρρ−−22ss3ρ+sρ3s−++s3ss−3)s,+) (26) froAmF(o3k2)kearn-Pdl(a3n4cak–beq).uaSteitotninfgorzP=j(xˆxˆ·t)ˆwiseroebatdailiyn found
wwehehraevse+ad=op(st1ed+Piesr2e)l/o√m2ova’nsdvasr−ia=ble(ss1γ−=is24)/α√2θ22, aanndd 0=−∂∂z[α(1+z2)+2jα(1−z2)−z(1+α2)]Pj(z)
ν = (1+α)2/4α. ThiscompletesStep2ofo−urprogram. + 21∂∂z22[(1−z2)(1−2αz+α2)]Pj(z), (35)
−
4
which has the unique properly normalized solution We recognize (43a) as the standard quantum limit
(SQL), but the expression (43b) for SqNfN is surprising.
1 α2 2j+2 We will call it the Hilbert correlation because it asserts
P (xˆ)= − . (36)
j 1 2αxˆ·tˆ+α2 that fN(t) is proportional to the Hilbert transform of
(cid:20) − (cid:21) qN(t). Physically, it ensures that fluctuations in the ob-
Substituting α = tanh1β per (25) yields Theorem 1. served position qM(t) are accompanied by force flucta-
− 4
This completes the third and final step of our proof tionssuchthatq(t)isattractedtowardqM(t),as(39a–b)
We now discuss some practical implications for exper- requires (and as is ubiquitious in spinometry).
iments inmagneticresonanceforcemicroscopy(MRFM) IstheHilbertcorrelationobservable? From(42)itcan
and gravity wave (GW) interferometry. We first write be shown that it is undetectable in qM(t)’s response to
the Ito equations (32) in Langevinform by substituting the classical force fext(t). Thus the standard quantum
limitforclassicalforcesignals,asdetectedinexperiments
δx tdt′x˙(t′) a(x ) r tdt′a(x(t′)) (37a)
n → 0 n → 0 likeGWinterferometry,isnotalteredbythepresence(or
Wn →rRgs 0tdt′xN(t′) b(xn)→bR(x(t)) (37b) abTsehnecea)uotfhaorHiislbperretsceonrtrlyelaitnivoens.tigating the conditions
where δt=1R/r is an interval and xN(t) is white noise underwhichHilbertcorrelationsare observable;inspino-
metric terms this seems to require a curved (nonlinear)
E[xNk(t)xNk′(t′)]=δkk′δ(t−t′)/gs2r. (38) dynamical manifold, as can arise, e.g., in MRFM when
several spins are present.
Then taking ∂/∂t, the resulting Langevin equation is
A major obstacle is that the differential geometry and
x˙ =rg2[a(x)+b(x)·(xM x)], (39a) topologyofdriftanddiffusionfunctionsonHilbertspaces
s − are poorly understood. The following would help:
where xM(t)=x(t)+xN(t) is the measured spin axis. Problem 1. Given a finite-dimensional Hilbert space
We see that x(t) is dynamically attracted toward the and on that space a set of Ito drift functions a(ψ ) and
measured axis xM(t). Even open-loop spinometers ex- diffusion functions b(ψ ), either exhibit a set o|f ispino-
hibit this attraction, since for α=0 we find meter operators A ,B| ithat generate a and b, or prove
i i
{ }
that no such operators exist.
x˙ =rg2[ 1x+ 1(I x x)·(xM x)]. (39b)
|α=0 s −4 2 − ⊗ − Whenstudentsask“HowdoestheStern-Gerlacheffect
Remark: a similar einselection-by-attraction is evident work?” or“Howdoesthestandardquantumlimitwork?”
evenin uniaxialspinometry,whereit dynamicallygener- the author increasingly answers them in terms of drift
ates the asymptotic “collapse” of ψ to an eigenstate, and diffusion functions. This focusses on the well-posed
n
| i
as described by (16) and Theorem 2. problem of designing these functions and realizing them
Wenowtransform(39a–b)tothesecond-orderNewto- inhardware,andyetmakesclearhowlittleisknown,and
nian equation of an oscillator. To do this, we introduce how much remains to be discovered.
a mass m and frequency ω by defining
0
qop =(~/jmω )1/2 [+s cos(ω t) s sin(ω t)], (40a)
0 2 0 1 0
−
pop =(mω0~/j)1/2 [−s2sin(ω0t)−s1cos(ω0t)]. (40b) ∗ Email:[email protected];
URL:http://courses.washington.edu/goodall/MRFM/;
We verify that for states with z 1 the canonical com-
mutator [qop,pop] = i~s /j i∼~ holds. Defining the Supported by the NIH/NCRR, the NSF/ECS, the Army
3 Research Office(ARO),and DARPA/DSO/MOSAIC.
≃
coherent oscillator coordinate q(t) to be [1] A.Perelomov,GeneralizedCoherent StatesandTheirAp-
plications (Springer-Verlag, 1986).
q(t)=(j~/mω0)1/2(y(t)cosω0t x(t)sinω0t), (41) [2] A.M.Perelomov,Commun.Mathe.Phys.26,222(1972).
− [3] J. M. Radcliffe, J. Phys. A 4, 313 (1971).
we find that (39b) takes the equivalent Newtonian form [4] K. Gottfried, Quantum Mechanics (W. A. Benjamin,
1966).
mq¨= mω2q+fext+fN qM =q+qN, (42) [5] M. E. Rose, Elementary Theory of Angular Momentum
− 0
(John Wiley & Sons, 1957).
where fext(t) is an arbitrary external force. From (38) [6] M. Abramowitz and I. A. Stegun, eds., Hndbk. of Mathe-
we obtain the noise spectral densities, which satisfy maticalFunctions (USGovernmentPrintingOffice,Wash-
ington, 1964), 10th ed.
SfNfN(ω)SqNqN(ω)= 41~2, (43a) [7] C19.8W5).,G2nadrdeinde.r,Hndbk.ofStochasticProcesses (Springer,
SqNfN(ω)= 21i~sgnω. (43b) [8] C. W. Gardiner and P. Zoller, Quantum Noise (Springer,
2000), 2nd ed.
Here SqNfN(ω)≡ −∞∞dτe−iωτE[qN(t)fN(t+τ)]. [9] W. H.Zurek, Rev.Mod. Phys. 75, 715 (2003).
R