Quantization of environment

Transcription

1 Quantization of environment Koji Usami Dated: September 8, 7 We have learned harmonic oscillators, coupled harmonic oscillators, and Bosonic fields, which are emerged as the contiuum limit N of the N coupled harmonics oscillators. With these ingredients we venture into the environmental problem of quantum mechanics. In particular we study an LCR circuit as an example. We learn how the enegy dissipation of LC circuit the system due to the resistor R the environment can be understood quantum mechanically by treating R as a semi-infinite D transmission line with characteristic impedance of Z p R. The model that the dissipative elements are treated as a collection of conservative reactive elements is called the Caldeira-Leggett model. The environment which is characterized by the frequency-independent impedance Z p is called Ohmic environment. I. LCR CIRCUIT - A CLASSICAL VIEW Let us start by discussing the LCR circuit classically. The circuit equation is given by the Langevin-type equation: The Fourier transform: gives L Qt + R Qt + C Qt Qt QΩ Vt. Noise voltage π QΩe iωt dtqωe iωt 3 Putting Ω L C Ω L QΩ irωqω + C QΩ VΩ. 4 we have the following algebraic equation: VΩ [ L Ω Ω iωr QΩ χ v ΩQΩ. 5 Here the susceptibility χ v Ω is intoduced. χ v Ω can be split into in-phase and quadarure parts, that is, χ v Ω χ vω + iχ vω with χ vω L Ω Ω 6 χ vω ΩR. 7 We can thus think that the noise voltage Vt at the resistor is induced as the response of the motion of the charge Qt. From Eq. 5 we get the spectral density of the charge S QQ Ω as S QQ Ω Q ΩQΩ χ v Ω V ΩVΩ χ v Ω SV V Ω, 8 Electronic address: usami@qc.rcast.u-tokyo.ac.jp

2 where S V V Ω is the spectral density of the voltage. Here S QQ Ω and S V V Ω are the single-sided spectral densities, which only assumes positive frequency Ω. The variance of Qt is obtained by integrating Eq. 8. To see how this connection arises let us invoke the Wiener-Khinchin therom: QτQ S QQ Ω Plugging τ in Eq. gives us the variance of Qt, that is, Qt Q Stationarity π S QQ Ω cos Ωτ 9 QτQ cos Ωτ, π S QQ Ω π π χ v Ω SV V Ω L Ω Ω + R L Ω S V V Ω When we are interested in the system in the thermal equilibrium the variance Qt can be obtained from the thermodynamical reasoning. Invoking the equipartition therem we have Qt C k BT. Now suppose that the spectral density of voltage is white, that is, S V V Ω S V V. By performing the integration in Eq. we have Qt S V V L π Ω Ω + R L Ω 4L Ω R S V V. 3 By comparing Eqs. and 3 we arrived at the well-known Nyquist formula: S V V 4R, 4 which shows the connection between the Ohmic dissipation R, the voltage fluctuation S V V, and the temperature. This can be recast into the following form R S V V Ω, 5 where we use the double-sided spectral density S V V Ω, which assumes potive and negative frequency. The single-sided spectral density S V V Ω can then be obtained in terms of them as S V V Ω S V V Ω + S V V Ω. 6 In classical setting we have S V V Ω S V V Ω, which can be derived from the fact that V tv V V t, that is, V t and V commute. II. QUANTIZING THE ENVIRONMENT [, A. Hamiltonian formalism Let us reexamine the LCR circuit from the viewpoint of Hamiltonian formalism hoping that we will gain more general tools to tackle open quantum systems. Here a difficulity is coming from the resistor R the environment in

3 the circuit, since for an harmonic oscillator the cannonical quantization is usually performed under the tacit assumption that the system is conservative. The basic idea to treat non-conservative LCR circuit quantum mechanically is to treat the dissipative element R as a semi-infinite D transmission line with characteristic impedance of Z p R i.e., the conservative element. Figure depicts a series LCR circuit a, which can be modeled as a series LC circuit capacitively coupled to a semi-infinite D transmission line b. 3 FIG. : a A series LCR circuit. b A series LC circuit capacitively coupled to a semi-infinite D transmission line. Let us see how the semi-infinite D transmission line produce the voltage noise at the junction to the series LC circuit. After taking the first and second continuum limits the Hamiltonian of a D transmission line can be expressed as H π hω k ĉ kĉk +, 7 where cωk ĉk h ĉ cωk k h φ k + φk i qk 8 cω k i q k, 9 cω k are the annihilation and creation operators in the node representation with the commutation relation Here the flux operator φk and the charge operator qk are given by respectively with the commutation relation: [ĉk, ĉ k πδk k. φk qk The Heisenberg equations of motion for ĉk and ĉ k are dxφxe ikx dxqxe ikx, [φk, qk i h πδk k. 3 ĉk, t ī h [H, ĉk, t iω k ĉk, t 4 ĉ k, t ī [ H, ĉ k, t iω k ĉ k, t 5 h

4 4 thus we have the plane wave solutions: With these results the charge variable qx, t is given by qx, t i ĉk, t ĉk, e iω kt 6 ĉ k, t ĉ k, e iωkt. 7 qk, teikx π hωk c π i π hωk c ĉ k, t ĉk, t e ikx ĉk, e ikx ωkt h.c., 8 which is indeed manifestly real as it has to be. The voltage Vx,t, which is also a real quantity, can be written in terms of qx, t as V x, t qx, t c hωk i ĉk, e ikx ωkt h.c.. 9 π c We now identify the modes with positive k as the right-moving modes and those with negative k as the left-moving modes. The right-moving voltage V x, t can thus be given by V x, t i i i where v p l lc is the velocity and Z p c can similarly given by hωk π c v p hω k π cv p hωzp dω π ĉk, e ikx ωkt h.c. ĉk, e ikx ωkt h.c. vp ĉωe ikx ωt h.c., 3 is the impedance of the D transmission line. The left-moving voltage V dω hωzp x, t i ĉωe ikx ωt h.c., 3 π where ĉω ĉk, vp, which satisfies the commutation relation: [ĉω, ĉ ω [ ĉk vp, ĉ k vp π v p δk k πδω ω. 3 While the average voltage fluctuation V x, t is zero under the thermal equilibrium the variance is not, which basically constitutes the Johnson-Nyquist noise. By evaluating the variance, or the spectral density S V V ω, we shall find the quantum version of the Nyquist formula. Let us consider the auto-correlation of the voltage at the open l terminal at x of a semi-infinite transmission line with the characteristic impedance Z p c, which can be given by V, t + τv, t V, t + τ + V, t + τ V, t + V, t 4 V, t + τv, t, 33 where refers to the average under the thermal equilibrium and we use V x, t V x, t + V x, t and V x, t V x, t, which are valid for the open terminal, for the first and second equalities, respectively.

5 For the situation in which the stationarity condition is satisfied the spectral density is obtained via the Wiener- Khinchin theorem: With Eq. 3 we have S V V Ω S V V Ω dω dω π π h Ω Z p dω hωz p 4 V, t + τv, t e iωτ hz p V, t + τv, t e iωτ V, t + τv, t e iωτ 4S V V Ω. 34 ωω ĉωcω e iω+ω t ĉωc ω e iω ω t e iω ωτ nω+πδω ω + ĉ ωcω e i ω+ω t + ĉ ωc ω e i ω ω t e iω+ωτ nωπδω ω dω hωz p nω + e iω ωτ + nωe iω+ωτ π nω + δω ω + nωδω + ω nω + ΘΩ + n Ω Θ Ω, 35 where Θx is the step function. Thus we have the voltage noise spectrum: S V V Ω 4S V V Ω h Ω Z p nω + ΘΩ + n Ω Θ Ω B. Anatomy of the Johnson-Nyquist noise Let us now take a step back and see what is going on here. For the real-valued classical variable V τ its autocorrelation function G V V τ V τv is also real. The commutativity of classical variable also suggests G V V τ G V V τ, that is, the auto-correlation is symmetric in time. This leads to the symmetric-in-frequency power spectrum: S V V Ω G V V τe iωτ G V V τ e iωτ S V V Ω. 37 G V V τ For the real-valued quantum variable V τ, however, its auto-correlation function G V V τ is not necessarily real! Let us see this in the following simple argument with a LC circuit. The real-valued flux variable is given by h ĉt φt + ĉ t hz ĉe iω t + ĉ e iω t, 38 C ω which is manifestly hermitian. The auto-correlation function is, however, not hermitian: G φφ τ φτφ hz hz ĉĉ e iωτ + ĉ ĉ e iωτ Thus we arrive the asymmetric-in-frequency power spectrum of the flux variable: S φφ Ω nω + e iω τ + nω e iω τ. 39 G φφ τe iωτ hz nω + πδω ω + nω πδω + ω. 4

6 6 SVV kbtzp ħ FIG. : Spectral density of quantum Johnson-Nyquist noise across a impedance matched to the characteristic impedance of the D transmission line, Z p. The blue dashed line shows the zero-temperature quantum noise while the red line shows the one at finite temperature. Since V t φt by the similar argument we have the asymmetric-in-frequency power spectrum: S V V Ω ω S φφ Ω hω Z nω + πδω ω + nω πδω + ω, 4 which is the discrete version of the spectral density in Eq. 35. Here we note that the dimensions of the prefactor in Eq. 35 and that in Eq. 4 coming from the dimension difference between ΘΩ and δω, i.e., [ and [ Ω. We see that the non-commutativity of the quantum operators φt and V t with those in different time and the emergent quantum fluctuation i.e., one extra photon in the positive frequency is the culprit of the asymmetric-in-frequency power spectrum. We also see that the result we have in Eq. 36 can be obtained by adding the contribution of infinitely many LC circuits with different frequencies. C. Quantum dissipation-fluctuation theorem The expression Eq. 36 can recast into more compact form: S V V Ω Z p hω + ΘΩ hω e hω k BT Z p hω ΘΩ + hω e hω Z p hω e hω Θ Ω k BT Θ Ω e hω e hω. 4 This is called a double-sided spectral density where the frequency Ω runs from negative to positive as shown in Fig.. Although the asymmetry of the spectral density in frequency is noticeable, it is not so trivial to see the asymmetry in practice. The reason is that the range of frequencies where the asymmetry is significant is Ω > kbt h, that is very high for the room temperature and GHz range even in a few mk environment. It is also required to distinguish between the positive and negative frequencies in order to see the asymmetry. Nevertheless there have been several experiments in which the quantum noise are revealed [3 8. The single-sided spectral density, which can be measured with standard spectrum analyzers, is, on the other hand,

7 given by symmetrizing the spectral density with respect to frequency: Z p hω Z p hω S V V Ω S V V Ω + S V V Ω + e hω e hω Z p hω coth hω 43 4Z p hω +, 44 e hω k BT where the frequency Ω runs only in the positive direction. In the last line we can recognize the contribution of the zero point fluctuation, Z p hω, to the noise spectral density explicitly. Equation 44 is called quantum dissipationfluctuation theorem, which connects the apparently unrelated two quantities; the transport coefficient Z p and the noise spectral density S V V Ω. By taking the classical limit hω the spectral density Eq. 44 becomes n hω S V V Ω 4Z p, 45 which is the well-known Johnson-Nyquist formula, where the spectrum is proportional to the impedance Z p and temperature T, which is indeed the same form as in Eq. 4 derived from the classical treatment of LCR circuit. The impedance Z p is, on the other hand, related to the difference of the noise spectral densities S V V Ω and S V V Ω; hω S V V Ω S V V Ω Z p hω Z p hω, 46 that is, e hω e hω Z p hω S V V Ω S V V Ω D. Ohmic environment We are thus able to treat a dissipative element characterized by the impedance Z p quantum mechanically. The quantum noise spectrum Eq. 4 shows peculiar quantum effect which manifest itself as the asymmetric-in-frequency power spectrum in the quantum regime hω. That the dissipative elements can be treated as a collection of conservative reactive elements is essentially the way in which the Caldeira-Leggett model deals with resisters quantum mechanically [,. The environment which is characterized by the frequency-independent impedance Z p and has the noise power spectrum Eq. 4 is called Ohmic environment. Here note that the quantum noise spectral density for the Ohmic environment Eq. 4 is propotional to Ω and is obtained within the assumption that the enviromment can be modeled as a +-dimensional Bosonic field. When the environment is modeled as D, 3D, or fractional dimension, the density of state ρω becomes f requency dependent for the Ohmic envionment ρω. Such environments exhibit either faster spacial dimension > or slower spacial dimension < development of noise power spectrum with respect to the frequency Ω than the Ohmic environment spacial dimension. Those are called super-ohmic and sub-ohmic environments, respectively. Appendix A: Relation to the linear-response theory [, 9, This kind of transport coefficient can in general be obtained by the linear-response theory. Let us check the above result can be reproduced from the linear-response theory. Suppose that the Hamiltonian H in Eq. 7 describes the unperturbed Hamiltonian for a D transmission line, which is now connected at x to an LC circuit capacitively with the intercation Hamiltonian H i ˆQ s t ˆV, A

8 where the canonical variables of the LC circuit are the charge ˆQ s t and the flux ˆΦ s t and ˆV ˆV x is Schrödinger s operator for the voltage at x see Eq. 9. In the interaction picture the time evolution of the dentiy operator for the D transmission line ρ I t is given by 8 where t ρ It ī h [H It, ρ I t, A H I t e i H t h H i e i H t h ˆQ s t e i H h t ˆV e i H h t ˆQ s t ˆV I t A3 and ρ I t e i H t h ρte i H h t A4 with ρt denoting the Schrödinger s dentiy operator. Assuming that the D transmission line is initially unperturbed and in the thermal equilibrium state ρ eq ρ ρ I. Then the formal solution of Eq. A can be obtained perturbatively as ρ I t ρ eq ī h ρ eq ī h ρ eq ī h ρ eq ī h ρ eq ī h where θt t is the step function: t t t dt [H I t, ρ I t dt [H I t, dt [H I t, ρ eq ρ eq ī h dt θt t [H I t, ρ eq t dt [H I t, ρ I t [ dt ˆQs t θt t ˆVI t, ρ eq, A5 θt t {, if t t, if t t <, used for extending the domain of integration to the infinity. Returning to the Schrödinger s density operator gives ρt ρ eq ī h ρ eq ī h [ dt ˆQs t θt t e i H h t t ˆV e i H h t t, ρ eq [ dt ˆQs t θt t ˆVI t t, ρ eq A6 A7 The expectation value of ˆV is then written as V t Tr [ρt ˆV s Tr [ρ eq ˆVs ī h ī h [[ dt ˆQs t θt t Tr ˆVI t t, ρ eq ˆVs [ dt ˆQs t θt t Tr ˆVs ˆVI t t ˆV I t t ˆV s ρ eq dt ˆQs t ī [ h θt t Tr ˆVs ˆVI t t ˆV I t t ˆV s ρ eq χ vt t dt ˆQs t χ v t t, A8

9 where we define the time-domain response function χ v τ as χ v τ ī h θτtr [ ˆVs ˆVI τ ˆV I τ ˆV s ρ eq ī h θτtr [ ˆVs e i H h τ ˆVs e i H h τ e i H h τ ˆVs e i H h τ ˆVs ρ eq ī h θτtr [ ˆVs e i H h τ ˆVs e i H h τ e i H h τ ˆVs e i H h τ ˆVs ρ eq ī [ h θτtr ˆVs e i H h τ ˆVs ρ eq e i H τ h e i H τ h ρ eq ˆVs e i H h τ ˆVs ī h θτtr e i H h τ ˆVs e i H h τ ˆV τ ˆV s ˆV ρ eq ρ eq ˆVs ˆV τ e i H h τ ˆVs e i H h ˆV τ ī h θτ ˆV τ ˆV ˆV ˆV τ. A9 The Fourier transform of Eq. A9 gives the susceptibility χ v Ω: 9 χ v Ω χ vω + iχ vω ī h ī h θτ ˆV τ ˆV ˆV ˆV τ e iωτ ˆV τ ˆV ˆV ˆV τ e iωτ. A The imaginary part χ vω can then read χ vω h [ Re [ h [ h h h h [ [ ˆV τ ˆV ˆV ˆV τ e iωτ ˆV τ ˆV ˆV ˆV τ e iωτ ˆV τ ˆV ˆV ˆV τ e iωτ + ˆV τ ˆV ˆV ˆV τ e iωτ ˆV τ ˆV ˆV ˆV τ e iωτ ˆV τ ˆV e iωτ h S V V Ω S V V Ω. According to Eq. 7, the impedance Z p can be given by + ˆV ˆV τ e iωτ ˆV τ ˆV ˆV ˆV τ e iωτ ˆV ˆV τ ˆV τ ˆV e iωτ ˆV ˆV τ ˆV τ ˆV e iωτ ˆV τ ˆV ˆV ˆV τ e iωτ A Z p χ vω Ω hω S V V Ω S V V Ω, A which agrees with Eq. 47. Here we assume the environment is in the thermal equilibrium and invoke the detailed balance condition: S V V Ω e hω k BT S V V Ω. Plugging this into Eq. 47 or A, the impedance becomes Z p e hω S V V Ω. hω A3 A4

10 In the classical limit T hω k B we have and reproduce the classical result obtained in Eq. 5. Z p S V V Ω, A5 Appendix B: Kubo formula Let us consider the situation in which a system parallel LC circuit and an environment semi-infinit D transmission line are inductively coupled see Fig. 3 as opposed to the situation we have investigated so far, where those are capacitively coupled see Fig.. The interaction Hamiltonian for the inductive coupling is given by H j ˆΦ s tît, B where Ît Îx, t is Schödinger s operator for the current at x, which can be wrtitten as Ît Z p V, t V, t σ p V, t V, t, B with σ p Z p B3 being the characteristic conductance of the D transmission line. The interaction Hamiltonian, Eq. B, for the inductively coupled system can be compared to the one, Eq. A, for the capcitively coupled system. Here V x, t and V x, t are defined in Eqs. 3 and 3, respectively. Let us here assume that V, t V, t as for the closed terminal at x. Using the similar argument developed in Sec. II, the current noise spectral density can be given by S II Ω IτI e iωτ σ p hω e hω. B4 Using the linear response theory developed in Sec. A with the interaction Hamiltonian H j in Eq. B, the expression for the conductance σ p can then be given in terms of S II Ω by which is called Kubo formula. σ p hω S IIΩ S II Ω, B5 [ M. H. Devoret, in Les Houches Session LXIII, Quantum Fluctuations, pp Elsevier, Amsterdam, 997. [ A. A. Clerk et al., Rev. Mod. Phys. 8, 55. [3 R. H. Koch, D. J. Van Harlingen, and J. Clarke, Phys. Rev. Lett. 47, [4 R. J. Schoelkopf, P. J. Burke, A. A. Kozhevnikov, D. E. Prober, and M. J. Rooks, Phys. Rev. Lett. 78, [5 R. Deblock, E. Onac, L. Gurevich, and L. P. Kouwenhoven, Science 3, 3 3. [6 K. Usami, Y. Nambu, B. -S. Shi, A. Tomita, and K. Nakamura, Phys. Rev. Lett. 9, [7 P. -M. Billangeon, F. Pierre, H. Bouchiat, and R. Deblock Phys. Rev. Lett. 96, [8 C. Riek, D. V. Seletskiy, A. S. Moskalenko, J. F. Schmidt, P. Krauspe, S. Eckart, S. Eggert, G. Burkard, and A. Leitenstorfer, Science 35, 4 5. [9 C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms John Wiley and Sons, New York, 989. [ C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions John Wiley and Sons, New York, 99.

11 FIG. 3: a A parallel LCR circuit. b A parallel LC circuit inductively coupled to a semi-infinite D transmission line.